Special Section Dedicated to Andre Vanderbauwhede

Shift dynamics near non-elementary T-points with real eigenvalues

Jürgen Knobloch Department of Mathematics, TU Ilmenau, Ilmenau, Germany.Correspondencejuergen.knobloch@tu-ilmenau.de
View further author information

Jeroen S. W. Lamb Department of Mathematics, Imperial College London, London, UK.View further author information

Kevin N. Webster Potsdam Institute For Climate Impact Research, Potsdam, Germany.View further author information

Special Section Dedicated to Andre Vanderbauwhede

Shift dynamics near non-elementary T-points with real eigenvalues

Abstract
Formulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

Abstract

We consider non-elementary T-points in reversible systems in $R^{2 n + 1}$ . We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Keywords:

Heteroclinic cycles reversible vector fields Lins’ method heteroclinic bifurcations

AMS Subject Classifications:

34C37 34C23 34C28 37G40 37G25

1. Introduction

T-points, sometimes also referred to as Bykov cycles, are heteroclinic cycles connecting two hyperbolic equilibria with different saddle indices. The heteroclinic connections $Γ_{1}$ and $Γ_{2}$ are assumed to be such that one of them, say $Γ_{1}$ , breaks up under perturbations while the other one is robust and isolated. The robustness of $Γ_{2}$ is due to the transversal intersection of the corresponding stable and unstable manifolds of the equilibria. We refer to the left sketch in Figure 1 for a visualisation.

These kind of cycle were first found in the Lorenz system [1]. Meanwhile T-points have been found to appear in many further applications such as Kuramoto–Sivashinsky systems, electronic oscillators, semiconductor lasers, magneto convection, and travelling waves in reaction–diffusion dynamics. For precise references concerning these applications we refer to [18].

Motivated by [1] Bykov studied in a series of papers the dynamics in a neighbourhood of those cycles, cf. e.g. [3] and references therein. More references can also be found in [14,18]. It turns out that the complexity of the nonwandering nearby dynamics depends to a large extent on the leading eigenvalues of the equilibria. If the leading eigenvalues of both equilibria are real then the corresponding dynamics is rather tame, whereas shift dynamics occurs in the presence of complex leading eigenvalues [3,18]. However, in [3] it is also claimed that double T-points give rise to shift dynamics also in the case of real leading eigenvalues. Here the notion double T-points refers to two T-points having the non-robust heteroclinic orbit $Γ_{1}$ in common. We refer to the right sketch in Figure 1 for a visualisation. Those double T-points will appear generically in the unfolding of a quadratic tangency of the corresponding manifolds $W^{u} (p_{1})$ and $W^{s} (p_{2})$ (notation is chosen according to Figure 1). In order to also capture those cycles we relax the notion T-point to the effect that we admit non-transversal intersections of the manifolds $W^{u} (p_{1})$ and $W^{s} (p_{2})$ along $Γ_{2}$ . We call the corresponding heteroclinic cycle degenerate T-point.

The aim of the paper is to study the dynamics in unfoldings of degenerate T-points. In doing so we demand the underlying vector field to be reversible with respect to a linear involution R. We show that Bykov’s result about shift dynamics near double T-points remains true within the reversible setting, cf. Theorem 1.1 below. Moreover we discuss the transition from shift dynamics to ‘no recurrent dynamics’. Our results suggest that this transition is mainly governed by subharmonic bifurcations from branches of periodic orbits.

In what follows we describe the precise setting and present our main results. Concluding this section we comment on the relation of our statements to existing results.

Figure 1. Sketch of a T-point (left) and of a double T-point (right) in $R^{3}$ in each case. The ‘classical’ notion T-point implies that the two-dimensional manifolds intersect transversely. Whereas we also allow non-transversal intersections.

Display full size

We consider a two-parameter family of vector fields $f : R^{2 n + 1} \times R^{2} \to R^{2 n + 1}$ ( $n \geq 1$ ), f smooth:(1.1) $\begin{matrix} \dot{x} = f (x, μ), μ = (μ_{1}, μ_{2}) . \end{matrix}$ (1.1)

We denote the flow of this vector field by ${ϕ_{μ}^{t}}$ . We assume that the family (1.1) is reversible with respect to a linear involution R, that is

(H1)

$R f (x, μ) = - f (R x, μ)$ ,

and we assume that the fixed point space of the involution R is n-dimensional

(H2)

$dim Fix R = n$ .

We refer to [15,25,31] or [6] for detailed information concerning reversible systems.

We aim to study the dynamics in the neighbourhood of a symmetric degenerate T-point.

For that we assume:

(H3)

At $μ = 0$ there exists a heteroclinic cycle $Γ$ between two hyperbolic equilibria of saddle type $p_{1}$ and $p_{2}$ with different saddle indices.

Without loss of generality we assume that

f (p_{i}, μ) = 0

i = 1, 2

for sufficiently small

| μ |

. This can be fixed with a (local) smooth change of coordinates.

The saddle index of an equilibrium is the dimension of the unstable manifold. Throughout this paper we denote the stable manifold of the equilibrium $p_{i}$ at parameter(s) $μ$ by $W^{s} (p_{i}, μ)$ . For brevity, we also denote $W^{s} (p_{i}, 0)$ by $W^{s} (p_{i})$ . In the same manner we use $W^{u} (p_{i}, μ)$ and $W^{u} (p_{i})$ to denote the corresponding unstable manifolds.

Let $\begin{matrix} Γ = Γ_{1} \cup Γ_{2}, \end{matrix}$

where $Γ_{1} = {q_{1} (t) : t \in R}$ denotes the heteroclinic solution connecting $p_{2}$ to $p_{1}$ , i.e. $Γ_{1} \subset W^{u} (p_{2}) \cap W^{s} (p_{1})$ , and similarly $Γ_{2} = {q_{2} (t) : t \in R}$ denotes the heteroclinic orbit connecting $p_{1}$ to $p_{2}$ , i.e. $Γ_{2} \subset W^{u} (p_{1}) \cap W^{s} (p_{2})$ . We also assume that

(H4)

$dim T_{q_{2} (0)} W^{s} (p_{2}) = dim T_{q_{2} (0)} W^{u} (p_{1}) = n + 1$ .

For the orbit

Γ_{1}

we assume

(H5)

$dim (T_{q_{1} (0)} W^{s} (p_{1}) \cap T_{q_{1} (0)} W^{u} (p_{2})) = 1$ .

Due to the last two hypotheses the heteroclinic orbit

Γ_{1}

will generically break up while moving

μ

away from zero. The heteroclinic orbit

Γ_{2}

however will generically be robust, or in other words, generically

W^{u} (p_{1})

and

W^{s} (p_{2})

will intersect transversely along

Γ_{2}

. In this case the cycle

Γ

is called a T-point, cf. [12]. Here however we assume that

W^{u} (p_{1})

and

W^{s} (p_{2})

have a further common tangent in addition to the vector field direction:

(H6)

$dim (T_{q_{2} (0)} W^{s} (p_{2}) \cap T_{q_{2} (0)} W^{u} (p_{1})) = 2$ .

This induces the notion degenerate T-point. Finally, the assumption that the heteroclinic cycle

Γ

is symmetric, that is

(H7)

$R Γ = Γ$ ,

justifies the notion symmetric degenerate T-point. Together with Hypothesis (H4) the symmetry of

Γ

implies that the fixed points

p_{1}

and

p_{2}

are non-symmetric but lie in the same group orbit, that is

R (p_{1}) = p_{2}

, and it implies that both

Γ_{1}

and

Γ_{2}

are symmetric heteroclinic orbits.

Figure 2. Sketch of an example of a non-elementary symmetric degenerate T-point heteroclinic cycle in $R^{3}$ between two real saddles.

Display full size

Hence we may assume that $\begin{matrix} q_{i} (0) \in Fix R, i \in {1, 2} . \end{matrix}$

Further, let $⟨ \cdot, \cdot ⟩$ be an R-invariant scalar product, cf. also Remark 2.6 below, and let(1.2) $\begin{matrix} Y_{i} : = {f (q_{i} (0), 0)}^{⊥}, i = 1, 2 . \end{matrix}$ (1.2)

With that we construct the cross-sections $Σ_{1}$ and $Σ_{2}$ as follows $\begin{matrix} Σ_{i} : = q_{i} (0) + Y_{i}, i = 1, 2 . \end{matrix}$

By construction we find that $Fix R \subset Y_{i}$ , $i = 1, 2$ . For the details regarding this as well as the following explanations we refer to Section 2. Further, Hypothesis (H6) gives rise to define $\begin{matrix} U : = T_{q_{2} (0)} W^{u} (p_{1}) \cap T_{q_{2} (0)} W^{s} (p_{2}) \cap Y_{2} . \end{matrix}$

According to Hypothesis (H6) the manifolds $W^{u} (p_{1})$ and $W^{s} (p_{2})$ have a common tangent along U in $Σ_{2}$ which, according to the symmetry of $Γ_{2}$ , belongs either to $Fix R$ or to $Fix (- R)$ . In the present paper we assume:

(H8)

The cycle $Γ$ is non-elementary, i.e.

	(i)	$U \subset Fix R$
	(ii)	$W^{s} (p_{2})$ and $W^{u} (p_{1})$ have a quadratic tangency along U.

The Assumption (H8)(ii) excludes further degeneracies which could be caused by a vanishing second order term. We refer to Figure 2 for a sketch of a T-point in

R^{3}

satisfying Hypotheses (H4)–(H8).

If alternatively it is assumed that $U \subset Fix (- R)$ , then the cycle $Γ$ is called elementary. The notions non-elementary and elementary are borrowed from the context of symmetric homoclinic orbits, cf. [15,31]. However, in this paper we only consider non-elementary T-points.

Under the above hypotheses both $Γ_{1}$ and $Γ_{2}$ have codimension one with respect to parameter unfoldings. We use the parameter $μ_{1}$ to unfold the splitting of $Γ_{1}$ , and we use the parameter $μ_{2}$ to unfold the splitting of $Γ_{2}$ . The first row in Figure 3 shows the unfolding of the quadratic tangency of $W^{u} (p_{1})$ and $W^{s} (p_{2})$ , cf. Hypothesis (H8)(ii). The situation depicted in the right panel (of this row) is referred to as a double T-point. By this we denote a situation where $W^{s} (p_{2})$ and $W^{u} (p_{1})$ intersect within two different isolated heteroclinic orbits directly connecting the equilibria $p_{1}$ and $p_{2}$ (without following $Γ_{1}$ ). We want to emphasise that here the double T-point appears in the unfolding of a degenerate T-point. Further we want to note that the representation in Figure 3 suggests that the quadratic tangency of the degenerate T-point can be unfolded by $μ_{2}$ . The corresponding justification is given in Section 2.1, where we also explain that the splitting of $W^{s} (p_{1})$ and $W^{u} (p_{2})$ can be controlled only by $μ_{1}$ .

Figure 3. Upper row: unfolding of the traces of $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ in $Σ_{2}$ corresponding to a non-elementary symmetric degenerate T-point heteroclinic cycle in $R^{3}$ . At $μ_{2} = 0$ the common tangency of $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ is parallel to $U \subset Fix R$ (in $R^{3}$ we have $U = Fix R$ ). Lower row: unfolding of the traces of $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ in $Σ_{2}$ corresponding to an elementary symmetric degenerate T-point heteroclinic cycle in $R^{3}$ . At $μ_{2} = 0$ the common tangency of $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ is parallel to $Fix (- R)$ .

Display full size

Furthermore, we assume that the equilibria $p_{1}$ and $p_{2}$ are real saddles. More precisely we assume:

(H9)

The leading stable and unstable eigenvalues $λ^{s}$ and $λ^{u}$ , respectively, of $p_{1}$ are real and simple.

The leading eigenvalues are the ones which are closest to the imaginary axis. Note that due to Hypotheses (H1) and (H4) the eigenvalues of

p_{2}

arise from the eigenvalues of

p_{1}

by multiplying with

(- 1)

. We assume

(H10)

$λ^{u} < | λ^{s} |$ .

In the same way as in [18] we make some further hypotheses ensuring that the T-point has codimension two:

(H11)

$Γ_{i} ⊄ W^{s s} (p_{i})$ , $Γ_{i} ⊄ W^{u u} (p_{i + 1})$ , $i = 1, 2$ .

Here

W^{s s} (p)

and

W^{u u} (p)

denotes the strong stable manifold and the strong unstable manifold, respectively, of the equilibrium p. This is a standard non-orbit flip condition. Furthermore, we assume a slight modification of the standard non-inclination flip condition for

Γ_{1}

. To this end we introduce the local extended-unstable manifold

W^{e u} (p_{2})

p_{2}

(this is an invariant manifold whose tangent space at

p_{2}

comprises the unstable and weakest stable directions), and correspondingly the extended-stable manifold

W^{e s} (p_{1})

p_{1}

. Note that these manifolds are not uniquely defined. However their tangent spaces along

q_{1}

are well defined. With this notation the non-inclination flip condition reads

(H12)

$W^{e u} {(p_{2})}_{q_{1} (0)} W^{e s} (p_{1})$ .

Our main result regarding the dynamics nearby

Γ

is the following:

Theorem 1.1:

Assume Hypotheses (H1)–(H12). There is a number $N_{s d} \in R^{+}$ and there is a constant $δ^{s} > 1$ such that for $μ = (μ_{1}, μ_{2})$ , $μ_{1} > 0$ and $μ_{2} > N_{s d} μ_{1}^{δ^{s}}$ the following applies: There is a neighbourhood of $Γ_{2} \cap Σ_{2}$ containing a set $S_{μ}$ which is invariant under the first-return-map $Π_{μ}$ (defined by the flow ${ϕ_{μ}^{t}}$ ), and $(S_{μ}, Π_{μ})$ is topologically conjugated to the full shift on two symbols. Moreover, $S_{μ}$ is R-invariant.

The signs of $μ_{i}$ , $i = 1, 2$ , are due to a sign condition regarding some coefficients in the bifurcation equation, see (3.3). To prove Theorem 1.1 we proceed in the spirit of [18]. To this end we adapt Lin’s method in such a way that we can handle the time-reversing symmetry and the common tangency of the manifolds $W^{u} (p_{1})$ and $W^{s} (p_{2})$ in $Σ_{2}$ . Roughly speaking the symbols are related to the two intersection points of $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ in $Σ_{2}$ , cf. right panel in the upper row in Figure 3. Note that for $μ_{1} = 0$ these intersection points correspond to non-degenerate symmetric T-points. According to results in [18] one may expect 1-periodic orbits in the unfolding of these T-points. Indeed, more specifically, the symbols for the shift dynamics are related to those periodic orbits.

Moreover we show that there is a subset $S_{μ, R}$ of $S_{μ}$ generating symmetric f-orbits. And we show that $(S_{μ, R}, Π_{μ})$ is topologically conjugated to a system which is chaotic in the sense of Devaney. In particular $S_{μ, R}$ contains all periodic orbits up to period six.

Recall that in the context of reversible vector fields an orbit $O$ is called symmetric if $R O = O$ . According to [31, Lemma 3] or [25, Theorem 4.1] the following holds true:

An orbit $γ$ is symmetric if and only if $O \cap Fix R \neq \emptyset$ .
A symmetric periodic orbit intersects $Fix R$ in exactly two different points.
A symmetric orbit which is not periodic intersects $Fix R$ in exactly one point.

We remark that although a general assumption in [31] demands an even dimensional phase space, the arguments in the proof of [31, Lemma 3] do not rely on this assumption. We also emphasise that according to [25, Theorem 4.3(iii)], symmetric periodic orbits are generically isolated in the present context, cf. also Hypothesis (H2).

Theorem 1.2:

Assume Hypotheses (H1)–(H12). Let $δ^{s} > 1$ be the constant according to Theorem 1.1. There is a number $N_{n r} \in R^{+}$ , $N_{n r} < N_{s d}$ such that for $μ = (μ_{1}, μ_{2})$ , $μ_{1} > 0$ and $μ_{2} < N_{n r} μ_{1}^{δ^{s}}$ there is no recurrent dynamics near $Γ_{1} \cup Γ_{2}$ .

Similar to the above, the signs of $μ_{i}$ , $i = 1, 2$ , are due to a sign condition (3.3).

Summarising, the statements of the foregoing theorems lead to the bifurcation diagram depicted in Figure 4.

Figure 4. Bifurcation diagram for the unfolding of a non-elementary T-point. (I): $μ$ -area for shift dynamics, (II): shift dynamics dissolves, (III): no recurrent dynamics near $Γ_{1} \cup Γ_{2}$ . The SD curve is given by $μ_{2} = N_{s d} μ_{1}^{δ^{s}}$ ( $μ_{1} > 0$ ), and the NR curve is given by $μ_{2} = N_{n r} μ_{1}^{δ^{s}}$ ( $μ_{1} > 0$ ).

Display full size

We say that a periodic orbit $O$ within a sufficiently small neighbourhood of $Γ$ has period $N \in N$ , or that $O$ is an N-periodic orbit, if $O$ follows the T-point cycle N times before closing the loop. In a preliminary study of the disappearance of the horseshoe in the region (II), we consider N-periodic orbits with $N \leq 4$ . Our study suggests that (for fixed $μ_{1}$ and increasing $μ_{2}$ ) all symmetric periodic orbits emerge either in the course of a saddle-center bifurcation or in the course of a subharmonic bifurcation. A more detailed bifurcation scenario is presented in Section 4. We note that with our tool to gain the corresponding bifurcation equations, Lin’s method, we are unable to make stability statements. However, due to the reversing symmetry there are no symmetric asymptotically stable periodic orbits, and so this excludes saddle-node bifurcations of those orbits.

Regarding one-periodic orbits near $Γ$ we have

Theorem 1.3:

Assume Hypotheses (H1)–(H12). Let $(μ_{1}, μ_{2})$ be within the region (II). There is a function $κ_{s c} (\cdot)$ such that for $μ_{2} > κ_{s c} (μ_{1})$ there are two one-periodic orbits. These two orbits merge at $μ_{2} = κ_{s c} (μ_{1})$ and disappear if $μ_{2}$ becomes smaller than $κ_{s c} (μ_{1})$ . All one-periodic orbits are symmetric.

Regarding two-periodic orbits near $Γ$ we have

Theorem 1.4:

Assume Hypotheses (H1)–(H12). Let $(μ_{1}, μ_{2})$ be within the region (II). There is a function $κ_{p d} (\cdot)$ such that for $μ_{2} > κ_{p d} (μ_{1})$ there is exactly one two-periodic orbit, which emerges from a one-periodic orbit in the course of a period doubling bifurcation at $μ_{2} = κ_{p d} (μ_{1})$ . For $μ_{2} < κ_{p d} (μ_{1})$ there is no two-periodic orbit.

Generically we have $κ_{p d} (μ_{1}) > κ_{s c} (μ_{1})$ . We comment on this fact in Section 3.2.

Theorem 1.5:

Assume Hypotheses (H1)–(H12). Let $(μ_{1}, μ_{2})$ be within the region (II). There is a function $κ_{3 s h} (\cdot)$ such that for $μ_{2} > κ_{3 s h} (μ_{1})$ there is a pair of symmetric 3-periodic orbits, which emerges from a one-periodic orbit in the course of a subharmonic bifurcation at $μ_{2} = κ_{3 s h} (μ_{1})$ . For $μ_{2} < κ_{3 s h} (μ_{1})$ there are no 3-periodic orbits.

Theorem 1.6:

Assume Hypotheses (H1)–(H12). Let $(μ_{1}, μ_{2})$ be within the region (II).

(i)	There is a function $κ_{4 s h} (\cdot)$ such that for $μ_{2} > κ_{4 s h} (μ_{1})$ there is a pair of symmetric 4-periodic orbits, which emerges from a one-periodic orbit in the course of a subharmonic bifurcation at $μ_{2} = κ_{4 s h} (μ_{1})$ .
(ii)	There is a function $κ_{2 p d} (\cdot)$ such that for $μ_{2} > κ_{2 p d} (μ_{1})$ there is a 4-periodic orbit, which emerges from the two-periodic orbit in the course of a period doubling bifurcation at $μ_{2} = κ_{2 p d} (μ_{1})$ .

In what follows we briefly discuss some related work. A lot of work has been done concerning the dynamics near T-points, see for example the references in [18]. Here we mention works that are related to the degeneracy of the T-point or to T-points in systems with an extra structure.

Bykov [3] claimed that in systems without any prescribed structure, such as time-reversibility, double T-points create a suspended horseshoe. Here we make a similar statement in the context of reversible systems, cf. Theorem 1.1. Furthermore, we study double T-points in the context of the unfolding of a degenerate T-point. In this unfolding we also study the disappearance of symmetric periodic orbits which are related to the shift dynamics, cf Theorems 1.3–1.6. As a result of this we present a tentative bifurcation diagram comprising symmetric orbits contained in $S_{μ, R}$ , see Figure 11.

Lamb et al. [26] studied symmetric T-points in reversible systems in $R^{3}$ . Besides their restriction to $R^{3}$ the main difference to the present work is that they considered non-degenerate T-points (where $Γ_{2}$ is robust). Also they assumed in contrast to our Assumption (H9) that $λ^{u}$ is complex. Their results are based on the study of an appropriate return map.

In the present paper we use Lin’s method, cf. [16,28] as the main tool for studying the nearby dynamics of $Γ$ . This method has proved to be a powerful tool in studying dynamics near T-points, cf. [18]. Here, however we need to adapt the framework given in [18] to the context of reversible systems. Simultaneously the additional common tangent direction U has to be incorporated into the method.

Labouriau and Rodrigues studied T-points in equivariant systems in a series of papers [19–21] and found very involved dynamics, also beyond shift dynamics. However, the setting they consider is very different from ours, for example they consider complex leading eigenvalues in contrast to our Assumption (H9).

In [10] Fernández-Sánchez et al. presented a model equation for a T-point in $R^{3}$ connecting two saddle-focus equilibria whose two-dimensional manifolds intersect non-transversely. The authors studied homoclinic orbits (to the same equilibrium) and show that these undergo a saddle-node bifurcation close to the T-point. It can be guessed that nearby periodic orbits undergo the same type of bifurcation. This effect, however can be seen as the non-reversible counterpart to the bifurcations of periodic orbits we described in Theorems 1.3–1.6. The paper [11] deals numerically with a similar object under $Z_{2}$ -symmetry.

In [9] reversible T-points are studied in $R^{3}$ . Assuming complex eigenvalues they relate the dynamics in an unfolding of the T-point to the Cocoon bifurcations; an accumulation of parameter values for which there exist heteroclinic tangencies between the equilibria.

The rest of the paper is organised as follows. In Section 2 we outline how Lin’s method can be adapted to the present setting. Central to this method are the so-called Lin orbits. These are sequences of partial orbits where jumps in certain directions are allowed between two consecutive partial orbits. We show how those orbits can be constructed, and how one can derive determination equations for actual orbits. Based on these determination equations we prove our main theorems in Section 3. Finally we discuss the aforementioned tentative bifurcation diagram for symmetric periodic orbits in Section 4.

2. Lin’s method

In this section we outline how Lin’s method can be adapted to the reversible setting. In doing so we assume throughout that the T-point $Γ$ is non-elementary, Hypothesis (H8).

At the core of Lin’s method are the so-called Lin orbits. In the context of T-points such orbits consist of pieces $X_{1, i}$ and $X_{2, i}$ of actual orbits, $X : = {(X_{1, i}, X_{2, i})}_{i \in Z}$ , see also [18]. The orbit piece $X_{1, i}$ starts in $Σ_{1}$ , follows $Γ_{1}$ until it reaches a neighbourhood of $p_{1}$ , then follows $Γ_{2}$ until it terminates in $Σ_{2}$ . Similarly the orbit piece $X_{2, i}$ starts in $Σ_{2}$ , follows $Γ_{2}$ until it reaches a neighbourhood of $p_{2}$ , then follows $Γ_{1}$ until it terminates in $Σ_{1}$ . Between two consecutive orbit pieces $X_{2, i}$ and $X_{1, i + 1}$ there may be a jump $Ξ_{1, i}$ in a particular direction $Z_{1} \subset Y_{1}$ . In addition, there may be a jump $Ξ_{2, i}$ in a particular direction $Z_{2} \subset Y_{2}$ between the two consecutive orbit pieces $X_{1, i}$ and $X_{2, i}$ . We refer to Figure 5 below for a visualisation.

Now, let $2 ω_{1, i}$ and $2 ω_{2, i}$ be (prescribed) transition times of $X_{1, i}$ and $X_{2, i}$ from $Σ_{1}$ to $Σ_{2}$ and $Σ_{2}$ to $Σ_{1}$ , respectively. For sufficiently large $ω_{j, i}$ we build sequences $\begin{matrix} ω : = {((ω_{1, i}, ω_{2, i}))}_{i \in Z} . \end{matrix}$

Furthermore, we consider sequences built of sufficiently small $u_{i} \in U$ $\begin{matrix} u : = {(u_{i})}_{i \in Z}, u_{i} \in U . \end{matrix}$

It can be proved that for each $μ$ which is sufficiently close to 0, and each such sequence $ω$ and $u$ , there exists a unique Lin orbit $X (u, ω, μ)$ , see Theorem 2.9 below.

By setting the jumps $Ξ_{j, i}$ ( $j = 1, 2$ , $i \in Z$ ) equal to zero one finds real orbits staying close to the heteroclinic cycle $Γ$ for all time. Therefore the bifurcation equation for orbits staying close to $Γ$ reads $\begin{matrix} Ξ : = (Ξ_{1, i} (u, ω, μ), Ξ_{2, i} (u, ω, μ))_{i \in Z} = 0 . \end{matrix}$

The subspace $Z_{1}$ is defined as follows:(2.1) $\begin{matrix} Z_{1} : = {(T_{q_{1} (0)} W^{s} (p_{1}) + T_{q_{1} (0)} W^{u} (p_{2}))}^{⊥}, \end{matrix}$ (2.1)

where again the orthogonality is with respect to the R-invariant inner-product $⟨ \cdot, \cdot ⟩$ . We further define $\begin{matrix} W_{1}^{+} = T_{q_{1} (0)} W^{s} (p_{1}) \cap Y_{1} and W_{1}^{-} = T_{q_{1} (0)} W^{u} (p_{2}) \cap Y_{1} . \end{matrix}$

Taking Hypothesis (H5) into consideration we have the following direct sum decomposition of $R^{2 n + 1}$ : $\begin{matrix} R^{2 n + 1} = span {f (q_{1} (0), 0)} \oplus W_{1}^{+} \oplus W_{1}^{-} \oplus Z_{1} . \end{matrix}$

Note that in accordance with Hypotheses (H4) and (H5) we find that $\begin{matrix} dim Z_{1} = 2 . \end{matrix}$

The following lemma states that R-invariant subspaces can be decomposed into subspaces of $Fix R$ and $Fix (- R)$ .

Lemma 2.1:

Let Z be an R-invariant subspace. Then $Z = (Z \cap Fix R) \oplus (Z \cap Fix (- R))$ .

ProofSince $Fix R \cap Fix (- R) = {0}$ it suffices to show that each element of $z \in Z$ can be written as a sum of elements of $Fix R$ and $Fix (- R)$ . The representation $z = (z + R z) + (z - R z)$ meets this condition.

So, taking the time-reversing symmetry of the vector field and the symmetry of $Γ_{1}$ into consideration we find

Lemma 2.2:

Assume (H1)–(H5) and (H7). Then the space $Z_{1}$ has a direct sum decomposition into one-dimensional subspaces of $Fix R$ and $Fix (- R)$ .

ProofDue to (H1) and $q_{1} (0) \in Fix R$ the vector field direction $f (q_{1} (0), 0)$ belongs to $Fix (- R)$ , and the spaces $W_{1}^{+}$ and $W_{1}^{-}$ are R-images of each other. So $W_{1}^{+} \oplus W_{1}^{-}$ is R-invariant and hence, according to Lemma 2.1, it has a direct sum decomposition into $(n - 1)$ -dimensional subspaces of $Fix R$ and $Fix (- R)$ , recall $dim W_{1}^{+} = dim W_{1}^{-} = n - 1$ . Since $Z_{1}$ is R-invariant it has a direct sum decomposition into subspaces of $Fix R$ and $Fix (- R)$ , cf. again Lemma 2.1. Now, counting dimensions gives the lemma.

To obtain corresponding statements related to $Γ_{2}$ we define in the same way as $Z_{1}$ $\begin{matrix} Z_{2} : = {(T_{q_{2} (0)} W^{s} (p_{2}) + T_{q_{2} (0)} W^{u} (p_{1}))}^{⊥} . \end{matrix}$

Note that in accordance with Hypotheses (H4) and (H6) we find that $\begin{matrix} dim Z_{2} = 1, \end{matrix}$

and further $\begin{matrix} W_{2}^{+} = (T_{q_{2} (0)} W^{s} (p_{2}) \cap Y_{2}) ⊖ U, W_{2}^{-} = (T_{q_{2} (0)} W^{u} (p_{1}) \cap Y_{2}) ⊖ U . \end{matrix}$

Again $⊖$ refers to a orthogonal decomposition. With that we find(2.2) $\begin{matrix} R^{2 n + 1} = span {f (q_{2} (0), 0)} \oplus W_{2}^{+} \oplus W_{2}^{-} \oplus U \oplus Z_{2} . \end{matrix}$ (2.2)

Note that by construction both U and its complement (in the direct sum decomposition (2.2)) are R-invariant.

Lemma 2.3:

Assume (H1)–(H8) then $Z_{2} \subset Fix (- R)$ .

ProofThe proof runs along the lines of the proof of Lemma 2.2.

2.1. Splitting of the stable and unstable Manifolds

The first step of Lin’s method is to study the splitting of the stable and unstable manifolds in $Σ_{i}$ as the parameter $μ$ is varied from zero. Neglecting the symmetry for the moment, the behaviour in $Σ_{1}$ is described by the following lemma.

Lemma 2.4:

[Lemma 3.1 in [18]] Assume (H4) and (H5). For each $μ$ which is sufficiently close to 0 there is a unique pair $(q_{1}^{+} (μ) (\cdot), (q_{1}^{-} (μ) (\cdot))$ of solutions of (1.1) such that:

(i)	$q_{1}^{+} (μ) (0) \in Σ_{1} \cap W^{s} (p_{1}, μ)$ , $q_{1}^{-} (μ) (0) \in Σ_{1} \cap W^{u} (p_{2}, μ)$ ,
(ii)	$\| q_{1}^{+} (μ) (t) - q_{1} (t) \|$ small $\forall t \in R^{+}$ and $\| q_{1}^{-} (μ) (t) - q_{1} (t) \|$ small $\forall t \in R^{-}$ ,
(iii)	$q_{1}^{+} (μ) (0) - q_{1}^{-} (μ) (0) \in Z_{1}$ .

For the proof of this lemma we refer to [18].

In $R^{3}$ , $n = 1$ the statement in $Σ_{1}$ is obvious. In this case both the stable manifold of $p_{1}$ and the unstable manifold of $p_{2}$ are one-dimensional, and the intersection with the two-dimensional hyperplane $Σ_{1}$ consists of single points in each case. The heteroclinic connection $Γ_{1}$ generally splits up under perturbation. Let $q_{1, μ}^{+}$ and $q_{1, μ}^{-}$ be determined by the ‘first hit’ of the stable manifold of $p_{1}$ and the unstable manifold of $p_{2}$ , respectively. Of course $q_{1, μ}^{+} - q_{1, μ}^{-} \in Z_{1}$ , recall that $Z_{1} = Y_{1}$ in this case. So, in a trivial way, for each $μ$ we find a unique pair of orbits in the stable manifold of $p_{1}$ and the unstable manifold of $p_{2}$ , respectively, such that the difference of their first hits in $Σ_{1}$ is in $Z_{1}$ .

Taking the time-reversing symmetry into consideration we additionally find

Lemma 2.5:

Assume (H1)–(H5) and (H7), and let $q_{1}^{\pm}$ be the (unique) partial orbits according to Lemma 2.4. Then

(i)	$R q_{1}^{+} (μ) (t) = q_{1}^{-} (μ) (- t)$ ,
(ii)	$q_{1}^{+} (μ) (0) - q_{1}^{-} (μ) (0) \in Fix (- R) \cap Z_{1}$ .

ProofThe pair $(R q_{1}^{-} (μ) (\cdot), R q_{1}^{+} (μ) (\cdot))$ also fulfils (i)–(iii) of Lemma 2.4. Therefore the uniqueness part of Lemma 2.4 provides $R q_{1}^{+} (μ) (t) = q_{1}^{-} (μ) (- t)$ and hence in particular $R q_{1}^{+} (μ) (0) = q_{1}^{-} (μ) (0)$ .

Based on the last two lemmas we define(2.3) $\begin{matrix} ξ_{1}^{\infty} (μ) : = q_{1}^{+} (μ) (0) - q_{1}^{-} (μ) (0) . \end{matrix}$ (2.3)

In $Z_{1}$ we introduce coordinates in a similar way as in [18, Section 4.1]. Here, however, we additionally take account of the reversing symmetry: According to Hypotheses (H1), (H7) and (H12) there are linearly independent $ζ_{1}^{1}$ , $ζ_{1}^{2} = R ζ_{1}^{1}$ such that $\begin{matrix} T_{q_{1} (0)} W^{e u} (p_{2}) \cap Y_{1} = W_{1}^{-} \oplus span {ζ_{1}^{2}}, T_{q_{1} (0)} W^{e s} (p_{1}) \cap Y_{1} = W_{1}^{+} \oplus span {ζ_{1}^{1}} . \end{matrix}$

Indeed, there is an R-invariant scalar product $⟨ \cdot, \cdot ⟩$ (see also Remark 2.6), which is in accordance with (1.2), and in respect of which we have(2.4) $\begin{matrix} ζ_{1}^{1} ⊥ ζ_{1}^{2}, W_{1}^{+} ⊥ W_{1}^{-} and ζ_{1}^{i} ⊥ W_{1}^{\pm}, i = 1, 2 . \end{matrix}$ (2.4)

That is $Z_{1} = span {ζ_{1}^{1}, ζ_{1}^{2}}$ , cf. (2.1). Furthermore, because of $ζ_{1}^{2} = R ζ_{1}^{1}$ we have(2.5) $\begin{matrix} ζ_{1}^{1} - ζ_{1}^{2} \in Fix (- R) . \end{matrix}$ (2.5)

Remark 2.6:

Note that by definition $Y_{1}$ is R-invariant, cf. (1.2). Now, let ${⟨ ⟨ \cdot, \cdot ⟩ ⟩}_{Y_{1}}$ be a scalar product on $Y_{1}$ such that $ζ_{1}^{1} ⊥ ζ_{1}^{2}$ , $W_{1}^{+} ⊥ W_{1}^{-}$ and $ζ_{1}^{i} ⊥ W_{1}^{\pm}$ , $i = 1, 2$ . Then, due to the symmetry of the involved quantities, ${⟨ \cdot, \cdot ⟩}_{Y_{1}} : = {⟨ ⟨ \cdot, \cdot ⟩ ⟩}_{Y_{1}} + {⟨ ⟨ R (\cdot), R (\cdot) ⟩ ⟩}_{Y_{1}}$ is an R-invariant scalar product on $Y_{1}$ such that (2.4) is valid with respect to ${⟨ \cdot, \cdot ⟩}_{Y_{1}}$ . Finally we define a scalar product $⟨ \cdot, \cdot ⟩$ on $R^{2 n + 1}$ by $⟨ \cdot |_{Y_{1}}, \cdot |_{Y_{1}} ⟩ : = {⟨ \cdot, \cdot ⟩}_{Y_{1}}$ and $Y_{1} ⊥ span {f (q_{1} (0), 0)}$ . Since $f (q_{1} (0), 0) \in Fix (- R)$ , this scalar product is R-invariant.

Furthermore, we assume that the splitting of $W^{u} (p_{2})$ and $W^{s} (p_{1})$ is only caused by $μ_{1}$ , recall from the introduction that $μ_{1}$ unfolds $Γ_{1}$ . So we get $\begin{matrix} D_{1} ξ_{1}^{\infty} (0) \neq 0 . \end{matrix}$

Together with Lemma 2.5 and (2.5) we may write $\begin{matrix} ξ_{1}^{\infty} (μ) = (\begin{matrix} μ_{1} \\ - μ_{1} \end{matrix}) . \end{matrix}$

Next we turn to the splitting $W^{u} (p_{1}, μ)$ and $W^{s} (p_{2}, μ)$ in $Σ_{2}$ . We find the following

Lemma 2.7:

Assume (H4) and (H6). For each pair $(u, μ)$ which is sufficiently close to (0, 0) there is a unique pair $(q_{2}^{+} (u, μ) (\cdot), (q_{2}^{-} (u, μ) (\cdot))$ of solutions of (1.1) such that:

(i)	$q_{2}^{+} (u, μ) (0) \in Σ_{2} \cap W^{s} (p_{2}, μ)$ , $q_{2}^{-} (u, μ) (0) \in Σ_{2} \cap W^{u} (p_{1}, μ)$ ,
(ii)	$\| q_{2}^{+} (u, μ) (t) - q_{2} (t) \|$ small $\forall t \in R^{+}$ and $\| q_{2}^{-} (u, μ) (t) - q_{2} (t) \|$ small $\forall t \in R^{-}$ ,
(iii)	$q_{2}^{+} (u, μ) (0) - q_{2}^{-} (u, μ) (0) \in Z_{2}$ ,
(iv)	$P_{U} (q_{2}^{\pm} (u, μ) (0)) = u$ .

Here

P_{U}

is the projection onto U related to the decomposition (2.2).

We note that due to the R-invariance of the direct sum decomposition (2.2) (cf. the explanation following (2.2)), we have(2.6) $\begin{matrix} R P_{U} = P_{U} R . \end{matrix}$ (2.6)

For the proof of Lemma 2.7 we refer to the proof of Lemma 1 in [15]. Similarly to (2.3) we define(2.7) $\begin{matrix} ξ_{2}^{\infty} (u, μ) : = q_{2}^{+} (u, μ) (0) - q_{2}^{-} (u, μ) (0) . \end{matrix}$ (2.7)

Similar to the considerations in $Σ_{1}$ we assume that the splitting of $W^{u} (p_{1})$ and $W^{s} (p_{2})$ is only caused by $μ_{2}$ , or in other words that $ξ_{2}^{\infty} (u, μ) = ξ_{2}^{\infty} (u, μ_{2})$ . In order to determine $ξ_{2}^{\infty}$ we have to incorporate Hypothesis (H8).

Lemma 2.8:

Assume (H1)–(H8). Then $R q_{2}^{+} (u, μ) (t) = q_{2}^{-} (u, μ) (- t)$ .

ProofThe lemma follows from the uniqueness statement and (iv) of Lemma 2.7 together with $U \subset Fix R$ .

Furthermore, the quadratic tangency translates into the following analytical expression related with $ξ_{2}^{\infty}$ , cf.: $\begin{matrix} {\frac{\partial (ξ_{2}^{\infty}, D_{1} ξ_{2}^{\infty})}{\partial (u, μ_{2})}|}_{(0, 0)} has rank two . \end{matrix}$

With that $ξ_{2}^{\infty}$ can be transformed into the simple form, cf. [15] or [17]. $\begin{matrix} ξ_{2}^{\infty} (u, μ) = μ_{2} - u^{2} . \end{matrix}$

We also refer to Figure 3. Note that in the depicted situation the jump $ξ_{2}^{\infty}$ is measured in the direction perpendicular to $Fix R$ .

2.2. Construction of Lin orbits

The next step in the method is to search for orbits $X_{j, i}$ , $j = 1, 2$ , $i \in Z$ , composing the Lin orbits $X = {(X_{1, i}, X_{2, i})}_{i \in Z}$ which we introduced at the beginning of Section 2.

We denote solutions of (1.1) by $x_{j, i} (\cdot)$ , corresponding to the orbits $X_{j, i}$ with $x_{j, i} (0) \in Σ_{j}$ and $x_{j, i} (2 ω_{j, i}) \in Σ_{j + 1}$ ; throughout the term ‘ $j + 1$ ’ is computed modulo 2. Actually $x_{1, i} (\cdot)$ is composed of solutions $x_{1, i}^{+} (\cdot)$ and $x_{2, i}^{-} (\cdot)$ which are defined on $[0, ω_{1, i}]$ and $[- ω_{1, i}, 0]$ , respectively. Similarly $x_{2, i} (\cdot)$ is composed of solutions $x_{2, i}^{+} (\cdot)$ and $x_{1, i}^{-} (\cdot)$ which are defined on $[0, ω_{2, i}]$ and $[- ω_{2, i}, 0]$ , respectively. This requires the coupling conditions(2.8) $\begin{matrix} x_{j, i}^{+} (ω_{j, i}) = x_{j + 1, i}^{-} (- ω_{j, i}), j = 1, 2, \end{matrix}$ (2.8)

and the jump conditions(2.9) $\begin{matrix} Ξ_{1, i} & : = x_{1, i + 1}^{+} (0) - x_{1, i}^{-} (0) \in Z_{1}, i \in Z, \end{matrix}$ (2.9) (2.10) $\begin{matrix} Ξ_{2, i} & : = x_{2, i}^{+} (0) - x_{2, i}^{-} (0) \in Z_{2}, i \in Z . \end{matrix}$ (2.10)

We refer to Figure 5 for a visualisation.

Figure 5. Ingredients of Lin orbits.

Display full size

Our main existence result in this respect is the following:

Theorem 2.9:

Assume Hypotheses (H1)–(H7). Then there are constants $c_{μ}$ , $c_{u}$ and $Ω$ such that for each $μ$ with $| μ | < c_{μ}$ , each $u$ with $| u_{i} | < c_{u}$ and each $ω$ with $ω_{j, i} > Ω$ , $j = 1, 2$ , $i \in Z$ , there is a unique sequence of solutions $x_{j, i}^{\pm} (u, ω, μ) (\cdot)$ , $j = 1, 2$ , $i \in Z$ , of (1.1) satisfying the coupling condition (2.8) and the jump condition (2.9), (2.10).

Moreover, these solutions satisfy $\begin{matrix} x_{2, i}^{+} (u, ω, μ) (0) - q_{2}^{+} (u_{i}, μ) (0) \in W_{2}^{+} \oplus W_{2}^{-} \oplus Z_{2} . \end{matrix}$

For the proof of this theorem we refer to [16].

In other words the theorem states that for the corresponding $(u, ω, μ)$ there is a Lin orbit $X (u, ω, μ) = {(X_{1, i}, X_{2, i})}_{i \in Z}$ , where the $X_{j, i}$ are related to the functions $x_{j, i}^{\pm}$ as described at the beginning of this section.

Furthermore, together with (2.10) and Lemma 2.7(iv) we find(2.11) $\begin{matrix} P_{U} (x_{2, i}^{\pm} (u, ω, μ) (0)) = u_{i} . \end{matrix}$ (2.11)

Based on Theorem 2.9 we define the following.

Definition 2.10:

Two Lin orbits $X (u, ω, μ) = {(X_{1, i}, X_{2, i})}_{i \in Z}$ and $\hat{X} (\hat{u}, \hat{ω}, μ) = {({\hat{X}}_{1, i}, {\hat{X}}_{2, i})}_{i \in Z}$ -pagination are equal, $X = \hat{X}$ , if there is an $i_{0} \in Z$ such that $X_{j, i} = {\hat{X}}_{j, i + i_{0}}$ , for $j = 1, 2$ and all $i \in Z$ .

Definition 2.11:

A Lin orbit $X (u, ω, μ) = {(X_{1, i}, X_{2, i})}_{i \in Z}$ is k-periodic, $k \in N$ , if for all $i \in Z$ we have $X_{j, i} = X_{j, i + k}$ .

Note that the latter definition implies that $x_{2, i}^{+} (u, ω, μ) (0) = x_{2, i + k}^{+} (u, ω, μ) (0)$ . Hence, due to (2.11), we have $u_{i} = u_{i + k}$ , i.e. $u$ is k-periodic. Furthermore, the definition implies that the domains of $X_{j, i}$ and $X_{j, i + k}$ coincide. This again means for the sequence $ω$ that $ω_{j, i} = ω_{j, i + k}$ , i.e. $ω$ is k-periodic. Altogether we find

Lemma 2.12:

A Lin orbit $X (u, ω, μ) = {(X_{1, i}, X_{2, i})}_{i \in Z}$ is k-periodic if and only if the sequences $u$ and $ω$ are k-periodic.

Lemma 2.13:

Let $X (u, ω, μ)$ be a Lin orbit. The R-image RX of X is a Lin orbit $\hat{X} (\hat{u}, \hat{ω}, μ)$ associated to the sequences $\hat{u} = {({\hat{u}}_{i})}_{i \in Z}$ , ${\hat{u}}_{i} = R u_{- i}$ and $\hat{ω} = {({\hat{ω}}_{1, i}, {\hat{ω}}_{2, i})}_{i \in Z}$ , ${\hat{ω}}_{j, i} = ω_{j + 1, - i}$ .

ProofWe define solutions ${\hat{x}}_{j, i}^{\pm} (\cdot)$ of (1.1) by(2.12) $\begin{matrix} {\hat{x}}_{j, i}^{+} (t) : = R x_{j, - i}^{-} (- t), {\hat{x}}_{j, i}^{-} (t) : = R x_{j, - i}^{+} (- t) . \end{matrix}$ (2.12)

The domains of ${\hat{x}}_{j, i}^{\pm}$ are strongly related to the domains of $x_{j, - i}^{\mp}$ . So, ${\hat{x}}_{1, i}^{+}$ , ${\hat{x}}_{2, i}^{+}$ , ${\hat{x}}_{1, i}^{-}$ and ${\hat{x}}_{2, i}^{-}$ are defined on $[0, ω_{2, - i}]$ , $[0, ω_{1, - i}]$ , $[- ω_{1, - i}, 0]$ and $[- ω_{2, - i}, 0]$ , respectively. With ${\hat{ω}}_{j, i} : = ω_{j + 1, - i}$ the functions ${\hat{x}}_{j, i}^{\pm} (\cdot)$ satisfy the coupling conditions (2.8) and also the jump conditions (2.9), (2.10). Also $P_{U} {\hat{x}}_{2, i}^{+} (0) = P_{U} R x_{2, - i}^{-} (0) = R P_{U} x_{2, - i}^{-} (0) = R u_{- i} = {\hat{u}}_{i}$ , cf. also (2.11) and (2.6). We denote the (partial) orbits connecting $Σ_{1}$ and $Σ_{2}$ corresponding to the solutions ${\hat{x}}_{j, i}^{\pm}$ (as constructed above – see also Figure 5) by ${\hat{X}}_{j, i}$ , $j = 1, 2$ , $i \in Z$ .

Altogether our construction shows that $\hat{X} = ({\hat{X}}_{1, i}, {\hat{X}}_{2, i})$ is a Lin orbit associated to $(\hat{u}, \hat{ω}, μ)$ . Since ${\hat{X}}_{j, i} = R X_{j + 1, - i}$ , $j = 1, 2$ , $i \in Z$ , we have $R X = \hat{X}$ .

Corollary 2.14:

Let X be a Lin orbit and let ${(Ξ_{1, i}, Ξ_{2, i})}_{i \in Z}$ and ${({\hat{Ξ}}_{1, i}, {\hat{Ξ}}_{2, 1})}_{i \in Z}$ be the sequences of jumps corresponding to X and RX, respectively. Then ${\hat{Ξ}}_{1, i} = - R Ξ_{1, - i - 1}$ and ${\hat{Ξ}}_{2, i} = - R Ξ_{2, - i}$ .

ProofAccording to (2.9), (2.10) and (2.12) we find $\begin{matrix} {\hat{Ξ}}_{1, i} & = {\hat{x}}_{1, i + 1}^{+} (0) - {\hat{x}}_{1, i}^{-} (0) = - R (x_{1, - i}^{+} (0) - x_{1, - i - 1}^{-} (0)) = - R Ξ_{1, - i - 1}, \\ {\hat{Ξ}}_{2, i} & = {\hat{x}}_{2, i}^{+} (0) - {\hat{x}}_{2, i}^{-} (0) = - R (x_{2, - i}^{+} (0) - x_{2, - i}^{-} (0)) = - R Ξ_{2, - i} . \end{matrix}$

According to Lemma 2.3 we find $Ξ_{2, i} \in Fix (- R)$ and hence ${\hat{Ξ}}_{2, i} = Ξ_{2, - i}$ .

Definition 2.15:

A Lin orbit X is symmetric if $X = R X$ .

Lemma 2.16:

A Lin orbit $X (u, ω, μ)$ is symmetric if and only if there is an $i_{0} \in Z$ such that(2.13) $\begin{matrix} ω_{j, i} = ω_{j + 1, - i - i_{0}}, and u_{i} = u_{- i - i_{0}}, i \in Z, j \in {1, 2} \end{matrix}$ (2.13)

ProofFirst we assume that X is symmetric, i.e. $X = R X = : \hat{X}$ . Then, according to Definition 2.10 there is an $i_{0} \in Z$ such that $X_{j, i} = {\hat{X}}_{j, i + i_{0}}$ . Furthermore, according to the considerations at the end of the proof of Lemma 2.13, we have ${\hat{X}}_{j, i + i_{0}} = R X_{j + 1, - i - i_{0}}$ . Hence(2.14) $\begin{matrix} X_{j, i} = R X_{j + 1, - i - i_{0}} . \end{matrix}$ (2.14)

This immediately gives $ω_{j, i} = ω_{j + 1, - i - i_{0}}$ . Also, according to (2.11) and (2.14) we find $\begin{matrix} u_{i} = P_{U} x_{2, i}^{+} (0) = P_{U} R x_{2, - i - i_{0}}^{-} (0) = u_{- i - i_{0}} . \end{matrix}$

Now, conversely we assume (2.13). Let $X (u, ω, μ)$ be the corresponding (unique) Lin orbit. The proof of Lemma 2.13 provides a representation of RX, and it turns out that $R X = X$ .

From Corollary 2.14 and Lemma 2.16 we get

Corollary 2.17:

Let $X (u, ω, μ)$ be a symmetric Lin orbit, and let $i_{0} \in Z$ be such that (2.13) is satisfied. Then $R Ξ_{1, i} = - Ξ_{1, - i - i_{0} - 1}$ and $R Ξ_{2, i} = - Ξ_{2, - i - i_{0}}$ .

2.3. The determination equation for orbits near $Γ$

A Lin orbit is a real orbit if all the jumps between two consecutive partial orbits are zero:(2.15) $\begin{matrix} Ξ_{j, i} (u, ω, μ) = 0, j = 1, 2, i \in Z . \end{matrix}$ (2.15)

With Theorem 2.9 we may write these jumps as $\begin{matrix} \begin{matrix} Ξ_{1, i} (u, ω, μ) & = x_{1, i + 1}^{+} (u, ω, μ) (0) - x_{1, i}^{-} (u, ω, μ) (0), \\ Ξ_{2, i} (u, ω, μ) & = x_{2, i}^{+} (u, ω, μ) (0) - x_{2, i}^{-} (u, ω, μ) (0) . \end{matrix} \end{matrix}$

As in [18, Section 3.2] we write(2.16) $\begin{matrix} x_{1, i}^{\pm} (u, ω, μ) (t) = q_{1}^{\pm} (μ) (t) + v_{1, i}^{\pm} (u, ω, μ) (t), \end{matrix}$ (2.16)

and correspondingly, cf. also Lemma 2.7,(2.17) $\begin{matrix} x_{2, i}^{\pm} (u, ω, μ) (t) = q_{2}^{\pm} (u_{i}, μ) (t) + v_{2, i}^{\pm} (u, ω, μ) (t) . \end{matrix}$ (2.17)

With that we find, cf. also (2.3) and (2.7), $\begin{matrix} \begin{matrix} Ξ_{1, i} (u, ω, μ) & = ξ_{1}^{\infty} (μ) + ξ_{1, i} (u, ω, μ), \\ Ξ_{2, i} (u, ω, μ) & = ξ_{2}^{\infty} (u_{i}, μ) + ξ_{2, i} (u, ω, μ), \end{matrix} \end{matrix}$

where(2.18) $\begin{matrix} \begin{matrix} ξ_{1, i} (u, ω, μ) & = v_{1, i + 1}^{+} (u, ω, μ) (0) - v_{1, i}^{-} (u, ω, μ) (0), \\ ξ_{2, i} (u, ω, μ) & = v_{2, i}^{+} (u, ω, μ) (0) - v_{2, i}^{-} (u, ω, μ) (0) . \end{matrix} \end{matrix}$ (2.18)

In order to obtain appropriate representations of $ξ_{j, i} (u, ω, μ)$ we follow the lines of [18, Section 4.1], while taking into consideration that in the present context some quantities depend on u. The following lemma can be seen as counterpart to [18, Lemma 4.4].

Lemma 2.18:

Assume Hypotheses (H1)–(H12). Then the jump $ξ_{1, i} (u, ω, μ)$ can be written in the form $\begin{matrix} ξ_{1, i} (u, ω, μ) & = (c_{1}^{1} (u_{i + 1}, μ) e^{- 2 λ^{u} ω_{1, i + 1}} + R_{1, i}^{1} (u, ω, μ)) ζ_{1}^{1} \\ + (c_{1}^{2} (u_{i}, μ) e^{- 2 λ^{u} ω_{2, i}} + R_{1, i}^{2} (u, ω, μ)) ζ_{1}^{2} \\ ξ_{2, i} (u, ω, μ) & = (c_{21} (u_{i}, μ) e^{2 λ^{s} ω_{1, i}} - c_{22} (u_{i}, μ) e^{2 λ^{s} ω_{2, i}} + R_{2, i} (u, ω, μ)) ζ_{2}, \end{matrix}$

where the quantities $c_{1}^{k}$ and $c_{2 k}$ , $k \in {1, 2}$ , are non-zero and $\begin{matrix} R_{1, i}^{1} (u, ω, μ) & = o (e^{- 2 λ^{u} ω_{1, i + 1}}) + o (e^{- 2 λ^{u} ω_{2, i}}), \\ R_{1, i}^{2} (u, ω, μ) & = o (e^{- 2 λ^{u} ω_{1, i + 1}}) + o (e^{- 2 λ^{u} ω_{2, i}}), \\ R_{2, i} (u, ω, μ) & = o (e^{2 λ^{s} ω_{1, i}}) + o (e^{2 λ^{s} ω_{2, i}}) . \end{matrix}$

Due to the eigenvalue condition (H10) it is enough to isolate the leading order terms containing $e^{- 2 λ^{u} ω}$ -terms. Furthermore, in the proof of [18, Lemma 4.4] the geometry of the T-point in $Σ_{2}$ (transversal intersection of $W^{u} (p_{1})$ and $W^{s} (p_{2})$ along $Γ_{2}$ ) was exploited at an essential point. For that reason we have to reconsider the proof in the present situation.

Preliminaries for the proof:

We start with introducing some notations and with specifying the setup.

The quantities $v_{j, i}^{\pm}$ introduced in (2.16) and (2.17) satisfy $\begin{matrix} {\dot{v}}_{1, i}^{\pm} = A_{1}^{\pm} (t, μ) v_{1, i}^{\pm} + g_{1}^{\pm} (t, v_{1, i}^{\pm}, μ) and {\dot{v}}_{2, i}^{\pm} = A_{2}^{\pm} (t, u_{i}, μ) v_{2, i}^{\pm} + g_{2}^{\pm} (t, v_{2, i}^{\pm}, u_{i}, μ) \end{matrix}$

where $\begin{matrix} A_{1}^{\pm} (t, μ) : = D_{x} f (q_{1}^{\pm} (μ) (t), μ) and A_{2}^{\pm} (t, u, μ) : = D_{x} f (q_{2}^{\pm} (u, μ) (t), μ) \end{matrix}$

and $\begin{matrix} g_{1}^{\pm} (t, v, μ) & : = f (q_{1}^{\pm} (μ) (t) + v, μ) - f (q_{1}^{\pm} (μ) (t), μ) - A_{1}^{\pm} (t, μ) v, \\ g_{2}^{\pm} (t, v, u, μ) & : = f (q_{2}^{\pm} (u, μ) (t) + v, μ) - f (q_{2}^{\pm} (u, μ) (t), μ) - A_{2}^{\pm} (t, u, μ) v, \end{matrix}$

respectively.

Let $Φ_{1}^{\pm} (μ, t, s)$ and $Φ_{2}^{\pm} (u, μ, t, s)$ be the transition matrices for the equations $\begin{matrix} \dot{v} = A_{1}^{\pm} (t, μ) v and \dot{v} = A_{2}^{\pm} (t, u, μ) v, \end{matrix}$

respectively. These equations have an exponential dichotomy on $R^{\pm}$ with corresponding projections $P_{1}^{\pm} (μ, t)$ , and $P_{2}^{\pm} (u, μ, t)$ , respectively. These projections commute with the transition matrix of the corresponding variational equation. Therefore they are determined by their image and by their kernel at $t = 0$ . Since the variational equations under consideration are related to solutions within a stable or an unstable manifold, respectively, the images of these projections coincides with the corresponding tangent spaces to these manifolds. We have $\begin{matrix} im P_{1}^{+} (μ, 0) = T_{q_{1}^{+} (μ) (0)} W^{s} (p_{1}), & im P_{1}^{-} (μ, 0) = T_{q_{1}^{-} (μ) (0)} W^{u} (p_{2}), \\ im P_{2}^{+} (u, μ, 0) = T_{q_{2}^{+} (u, μ) (0)} W^{s} (p_{2}), & im P_{2}^{-} (u, μ, 0) = T_{q_{2}^{-} (u, μ) (0)} W^{u} (p_{1}) . \end{matrix}$

Regarding the kernels of the projections at $t = 0$ we stipulate $\begin{matrix} ker P_{1}^{+} (μ, 0) = W_{1}^{-} \oplus Z_{1}, & ker P_{1}^{-} (μ, 0) = W_{1}^{+} \oplus Z_{1}, \\ ker P_{2}^{+} (u, μ, 0) = W_{2}^{-} \oplus Z_{2}, & ker P_{2}^{-} (u, μ, 0) = W_{2}^{+} \oplus Z_{2} . \end{matrix}$

In this respect we also introduce quantities $a_{j, i}^{\pm}$ : $\begin{matrix} (I - P_{1}^{+} (μ, ω_{1, i})) v_{1, i}^{+} (ω_{1, i}) = a_{1, i}^{+}, & (I - P_{1}^{-} (μ, - ω_{2, i})) v_{1, i}^{-} (- ω_{2, i}) = a_{2, i}^{-}, \\ (I - P_{2}^{+} (u_{i}, μ, ω_{2, i})) v_{2, i}^{+} (ω_{2, i}) = a_{2, i}^{+}, & (I - P_{2}^{-} (u_{i}, μ, - ω_{1, i})) v_{2, i}^{-} (- ω_{1, i}) = a_{1, i}^{-}, \end{matrix}$

Furthermore, we denote the transition matrix of the adjoint of the variational equation along $q_{1}^{\pm} (μ) (\cdot)$ by $Ψ_{1}^{\pm} (μ, \cdot, \cdot)$ , and correspondingly we denote the transition matrix of the adjoint of the variational equation along $q_{2}^{\pm} (u, μ) (\cdot)$ by $Ψ_{2}^{\pm} (u, μ, \cdot, \cdot)$ .

For our analysis in the proof we assume the following setup.

We start with specifying extended unstable manifolds of $p_{2}$ with respect to the vector field $f (\cdot, μ)$ : Note that $W_{2}^{-} \oplus Z_{2}$ is transversal to the stable manifold of $p_{2}$ , cf. also (2.2). Then (due to the inclination lemma) the image of $q_{2}^{+} (0, μ) (0) + (W_{2}^{-} \oplus Z_{2})$ under the flow belongs to an extended unstable manifold of $p_{2}$ . We stipulate, cf. also Figure 6, $\begin{matrix} W^{e u} (p_{2}, μ) : {ϕ_{μ}^{t} (q_{2}^{+} (0, μ) (0) + (W_{2}^{-} \oplus Z_{2})) : t \geq 0} \subset W^{e u} (p_{2}, μ) . \end{matrix}$

Accordingly we specify an extended stable manifold of $p_{1}$ by $\begin{matrix} W^{e s} (p_{1}, μ) : = R (W^{e u} (p_{2}, μ)) . \end{matrix}$

Hence $\begin{matrix} {ϕ_{μ}^{t} (q_{2}^{-} (0, μ) (0) + (W_{2}^{+} \oplus Z_{2})) : t \leq 0} \subset W^{e s} (p_{1}, μ) . \end{matrix}$

In Remark 2.19 below we show that there are transformations respecting the reversing symmetry and realising the following assumptions.

(A1)

Let $V_{q_{1} (0)}$ be a neighbourhood of $q_{1} (0)$ . For small $| μ |$ it is true:

\begin{matrix} W^{e u} (p_{2}, μ) \cap V_{q_{1} (0)} \supset & q_{1}^{-} (μ) (0) + (W_{1}^{-} \oplus span {ζ_{1}^{2}}), \\ W^{e s} (p_{1}, μ) \cap V_{q_{1} (0)} \supset & q_{1}^{+} (μ) (0) + (W_{1}^{+} \oplus span {ζ_{1}^{1}}) . \end{matrix}

(A2)

Let $V_{p_{2}}$ be a neighbourhood of $p_{2}$ . For small $| μ |$ it is true:

\begin{matrix} W^{e u} (p_{2}, μ) \cap V_{p_{2}} \subset T_{p_{2}} W^{e u} (p_{2}, μ), W^{s} (p_{2}, μ) \cap V_{p_{2}} \subset T_{p_{2}} W^{s} (p_{2}, μ) . \end{matrix}

Due to the reversing symmetry for small $| μ |$ it is then also true that within a sufficiently small neighbourhood $V_{p_{1}}$ of $p_{1}$ $\begin{matrix} W^{e s} (p_{1}, μ) \cap V_{p_{1}} \subset T_{p_{1}} W^{e s} (p_{1}, μ), W^{u} (p_{1}, μ) \cap V_{p_{1}} \subset T_{p_{1}} W^{u} (p_{1}, μ) . \end{matrix}$

The Assumptions (A1) and (A2) entail the following, we refer to Figure 6,(2.19) $\begin{matrix} Ψ_{1}^{-} (μ, - ω, 0) ζ_{1}^{1} ⊥ T_{p_{2}} W^{e u} (p_{2}, μ) \end{matrix}$ (2.19)

and(2.20) $\begin{matrix} Ψ_{1}^{+} (μ, ω, 0) ζ_{1}^{1} \in W_{l o c}^{e s} (p_{1}, μ), Ψ_{1}^{+} (μ, ω, 0) ζ_{1}^{1} ⊥ W^{s} (p_{1}, μ) . \end{matrix}$ (2.20)

The latter properties imply(2.21) $\begin{matrix} ⟨ Ψ_{1}^{+} (μ, ω, 0) ζ_{1}^{1}, e_{p_{1}}^{u} ⟩ \neq 0, \end{matrix}$ (2.21)

where $e_{p_{1}}^{s}$ denotes the leading stable eigendirection of $p_{1}$ .

Figure 6. Sketch of the consequences (2.19), cf. left panel, and (2.21), cf. right panel, of the Assumptions (A1) and (A2).

Display full size

Remark 2.19:

In what follows we sketch transformations realising the Assumptions (A1) and (A2). These transformations are local ones by nature, but can be globalised by means of appropriate cut-off functions. In order to preserve the reversibility of the vector field the (local) transformation near $q_{1} (0)$ must respect the reversing symmetry (this transformation must commute with R) and the corresponding cut-off function must be R-invariant. Let $T_{p_{2}}$ be the transformation near $p_{2}$ realising (A2), then $T_{p_{1}} : = R \circ T_{p_{2}} \circ R$ is the corresponding transformation realising the counterpart of (A2) at $p_{1}$ . Applying both $T_{p_{1}}$ and $T_{p_{2}}$ preserves the reversible structure.

Transformation near $q_{1} (0)$ : In a first step we straighten the intersection of $W^{e u} (p_{2}, μ)$ and $W^{e s} (p_{1}, μ) = R (W^{e u} (p_{2}, μ))$ . Note that this intersection consists of a symmetric f-orbit. Hence $W^{e s} (p_{1}, μ) \cap W^{e u} (p_{2}, μ)$ intersects $Fix R$ in exactly one point. $W^{e s} (p_{1}, μ) \cap W^{e u} (p_{2}, μ) \cap V_{1}$ can be written as $\begin{matrix} W^{e s} (p_{1}, μ) \cap W^{e u} (p_{2}, μ) \cap V_{1} = q_{1} (0) + graph h^{s u}, \end{matrix}$ $h^{s u} : span {f (q_{1} (0), 0)} \to W_{1}^{+} \oplus W_{1}^{-} \oplus Z_{1}$ . Note that, due to the symmetry $h^{s u}$ commutes with R. Furthermore, write $x \in span {f (q_{1} (0), 0)} \oplus W_{1}^{+} \oplus W_{1}^{-} \oplus Z_{1}$ in the form $x = x^{f} + x^{r}$ , $x^{f} \in span {f (q_{1} (0), 0)}$ and $x^{r} \in W_{1}^{+} \oplus W_{1}^{-} \oplus Z_{1}$ . Define a transformation $T^{s u}$ in $span {f (q_{1} (0), 0)} \oplus W_{1}^{+} \oplus W_{1}^{-} \oplus Z_{1}$ by $T^{s u} (x) : = x^{f} + (x^{r} + h^{s u} (x^{f}))$ . Note that $T^{s u} (x^{f})$ lies on the graph of $h^{s u}$ . Hence, $W^{e s} (p_{1}, μ) \cap W^{e u} (p_{2}, μ) \cap V_{1}$ can be straightened by means of the inverse of $T^{s u}$ . Finally, since $h^{s u}$ commutes with R also $T^{s u}$ commutes with R. That is, $(T^{s u})^{- 1}$ preserves the reversible structure.

In the next step we assume that $W^{e s} (p_{1}, μ) \cap W^{e u} (p_{2}, μ) \cap V_{1}$ is already straightened. For $x \in span {f (q_{1} (0), 0)} \oplus W_{1}^{+} \oplus span {ζ_{1}^{1}} \oplus W_{1}^{-} \oplus span {ζ_{1}^{2}}$ we introduce coordinates $(x^{f}, x^{+}, x^{1}, x^{-}, x^{2})$ and identify $q_{1} (0) + x \in V_{1}$ with $(x^{f}, x^{+}, x^{1}, x^{-}, x^{2})$ . Within this setting we have $\begin{matrix} W^{e u} (p_{2}) \cap V_{1} & = \{(x^{f}, (x^{+}, x^{1}) + h^{e u} (x^{f}, x^{-}, x^{2}), x^{-}, x^{2})\}, \\ W^{e s} (p_{1}) \cap V_{1} & = \{(x^{f}, x^{+}, x^{1}, (x^{-}, x^{2}) + h^{e s} (x^{f}, x^{+}, x^{1}))\}, \end{matrix}$

where $\begin{matrix} h^{e u} & : span {f (q_{1} (0), 0)} \oplus W_{1}^{-} \oplus span {ζ_{1}^{2}} \to W_{1}^{+} \oplus span {ζ_{1}^{1}}, \\ h^{e s} & : span {f (q_{1} (0), 0)} \oplus W_{1}^{+} \oplus span {ζ_{1}^{1}} \to W_{1}^{-} \oplus span {ζ_{1}^{2}} . \end{matrix}$

Note that $\begin{matrix} h^{e u} (x^{f}, 0, 0) = 0 and h^{e s} (x^{f}, 0, 0) = 0 \end{matrix}$

and $\begin{matrix} R h^{e u} (x^{f}, x^{-}, x^{2}) = h^{e s} (R (x^{f}, x^{-}, x^{2})) . \end{matrix}$

Now we define a transformation $\begin{matrix} T : (x^{f}, x^{+}, x^{1}, x^{-}, x^{2}) \mapsto (x^{f}, (x^{+}, x^{1}) + h^{e u} (x^{f}, x^{-}, x^{2}), (x^{-}, x^{2}) + h^{e s} (x^{f}, x^{+}, x^{1})) . \end{matrix}$

Note that $T$ maps $(x^{f}, 0, 0, x^{-}, x^{2})$ on the graph of $h^{e u}$ and $(x^{f}, x^{+}, x^{1}, 0, 0)$ on the graph of $h^{e s}$ . Hence $T^{- 1}$ flattens simultaneously $W^{e u} (p_{2}, μ)$ and $W^{e s} (p_{1}, μ)$ . Finally, $T$ commutes with R.

Altogether $T^{- 1} \circ (T^{s u})^{- 1}$ realises (A1).

Transformation near $p_{2}$ : In principle we can proceed as for the transformation near $q_{1} (0)$ . In this respect we note that $W^{e u} (p_{2}, μ)$ and $W^{s} (p_{2}, μ)$ intersect along a curve (consisting of three orbits, namely the equilibrium $p_{2}$ and two orbits within the stable manifold. In comparison with the previous case we only have to replace the f-orbit by this curve. Recall that in this case we do not have to regard the reversing symmetry.

Proof of Lemma 2.18:

To analyse the jump $ξ_{1, i} (u, ω, μ)$ we use the following representation, see also (2.18): $\begin{matrix} ξ_{1, i} (u, ω, μ) & = \sum_{j = 1}^{2} ⟨ ζ_{1}^{j}, ξ_{1, i} (u, ω, μ) ⟩ ζ_{1}^{j} \\ = \sum_{j = 1}^{2} (〈 ζ_{1}^{j}, (I - P_{1}^{+} (μ, 0)) v_{1, i + 1}^{+} (u, ω, μ) (0) \\ - (I - P_{1}^{-} (μ, 0)) v_{1, i}^{-} (u, ω, μ) (0) 〉) ζ_{1}^{j} . \end{matrix}$

First we consider the term $〈 ζ_{1}^{1}, (I - P_{1}^{+} (μ, 0)) v_{1, i + 1}^{+} (u, ω, μ) (0) 〉$ . By standard theory [16,28] we find, cf. also [18, Eqn. (4.6)] or [13, Section 3.3], $\begin{matrix} 〈 ζ_{1}^{1}, (I - P_{1}^{+} (μ, 0)) v_{1, i + 1}^{+} (u, ω, μ) (0) 〉 \\ = 〈 Ψ_{1}^{+} (μ, ω_{1, i + 1}, 0) {(I - P_{1}^{+} (μ, 0))}^{*} ζ_{1}^{1}, P_{1} (u_{i + 1}, μ, ω_{1, i + 1}) q_{2}^{-} (u_{i + 1}, μ) (- ω_{1, i + 1}) 〉 \\ + o (e^{- 2 λ^{u} ω_{1, i + 1}}), \end{matrix}$

where $P_{1} (u, μ, t)$ is a projection projecting on $im (I - P_{1}^{+} (μ, t))$ along $im (I - P_{2}^{-} (u, μ, - t))$ .

Next we consider the terms in the scalar product on the right-hand side in the last equation. Due to Hypothesis (H11) the solution $q_{2}^{-} (u, μ) (t)$ behaves asymptotically as $t \to - \infty$ like $e^{λ^{u} t} η_{1}^{u} (u, μ)$ , where $η_{1}^{u} (u, μ) \neq 0$ is parallel to the leading unstable eigendirection of $p_{1}$ which is spanned by $e_{p_{1}}^{u}$ . The asymptotical behaviour of $q_{2}^{-}$ becomes clear by [13, Lemma 3.3]. More precisely we find that(2.22) $\begin{matrix} P_{1} (u, μ, t) q_{2}^{-} (u, μ) (t) = e^{λ^{u} t} η_{1}^{u} (u, μ) + o (e^{λ^{u} t}), as t \to - \infty . \end{matrix}$ (2.22)

We remark that asymptotically, as $t \to \infty$ , $P_{1} (u, μ, t)$ tends to the projection which maps on the unstable subspace of $p_{1}$ along the stable subspace of $p_{1}$ . Precise estimates of $P_{1} (u, μ, t)$ which finally ensure (2.22) can be found in [28] or [16], respectively. Furthermore, $Ψ_{1}^{+} (μ, ω_{1, i + 1}, 0) {(I - P_{1}^{+} (μ, 0))}^{*} ζ_{1}^{1}$ behaves asymptotically like $e^{- λ^{u} t} {\hat{η}}_{1}^{s} (μ)$ as $t \to \infty$ , where ${\hat{η}}_{1}^{s} (μ) \neq 0$ . The asymptotical behaviour becomes clear by our construction, cf. (2.20) and [13, Lemma 3.4]. More precisely we find that(2.23) $\begin{matrix} Ψ_{1}^{+} (μ, t, 0) {(I - P_{1}^{+} (μ, 0))}^{*} ζ_{1}^{1} = e^{- λ^{u} t} {\hat{η}}_{1}^{s} (μ) + o (e^{- λ^{u} t}), as t \to \infty . \end{matrix}$ (2.23)

Altogether this yields $\begin{matrix} 〈 ζ_{1}^{1}, (I - P_{1}^{+} (μ, 0)) v_{1, i + 1}^{+} (u, ω, μ) (0) 〉 = 〈 {\hat{η}}_{1}^{s} (μ), η_{1}^{u} (u_{i + 1}, μ) 〉 e^{- 2 λ^{u} ω_{1, i + 1}} + o (e^{- 2 λ^{u} ω_{1, i + 1}}) . \end{matrix}$

According to (2.21) we find(2.24) $\begin{matrix} c_{1}^{1} (u_{i + 1}, μ) : = 〈 {\hat{η}}_{1}^{s} (μ), η_{1}^{u} (u_{i + 1}, μ) 〉 \neq 0 . \end{matrix}$ (2.24)

Now we turn to the term $〈 ζ_{1}^{1}, - (I - P_{1}^{-} (μ, 0)) v_{1, i}^{-} (u, ω, μ) (0) 〉$ . Again by standard theory we find(2.25) $\begin{matrix} 〈 ζ_{1}^{1}, - (I - P_{1}^{-} (μ, 0)) v_{1, i}^{-} (u, ω, μ) (0) 〉 \\ = 〈 Ψ_{1}^{-} (μ, - ω_{2, i}, 0) {(I - P_{1}^{-} (μ, 0))}^{*} ζ_{1}^{1}, P_{2} (u_{i}, μ, ω_{2, i}) q_{2}^{+} (u_{i}, μ) (- ω_{2, i}) 〉 + o (e^{- 2 λ^{u} ω_{2, i}}), \end{matrix}$ (2.25)

where $P_{2} (u, μ, t)$ is a projection projecting on $im (I - P_{1}^{-} (μ, - t))$ along $im (I - P_{2}^{+} (u, μ, t))$ .

In quite the same way as above we find as a counterpart of (2.22)(2.26) $\begin{matrix} P_{2} (u, μ, t) q_{2}^{+} (u, μ) (t) = e^{- λ^{u} t} η_{2}^{s} (u, μ) + o (e^{- λ^{u} t}), as t \to \infty, \end{matrix}$ (2.26)

where $η_{2}^{s} (u, μ)$ is parallel to the leading stable direction of $p_{2}$ . For the equivalent of (2.23) it is enough to take down(2.27) $\begin{matrix} Ψ_{1}^{-} (μ, t, 0) {(I - P_{1}^{-} (μ, 0))}^{*} ζ_{1}^{1} = O (e^{λ^{u} t}), as t \to - \infty . \end{matrix}$ (2.27)

Plugging in into (2.25) yields $\begin{matrix} 〈 ζ_{1}^{1}, - (I - P_{1}^{-} (μ, 0)) v_{1, i}^{-} (u, ω, μ) (0) 〉 \\ = 〈 Ψ_{1}^{-} (μ, - ω_{2, i}, 0) {(I - P_{1}^{-} (μ, 0))}^{*} ζ_{1}^{1}, η_{2}^{s} (u, μ) 〉 e^{- λ^{u} ω_{2, i}} + o (e^{- 2 λ^{u} ω_{2, i}}) . \end{matrix}$

Due to (2.19) and the definition of the projection $P_{1}^{-}$ we have (see also the left panel in Figure 6),(2.28) $\begin{matrix} 〈 Ψ_{1}^{-} (μ, - ω_{2, i}, 0) {(I - P_{1}^{-} (μ, 0))}^{*} ζ_{1}^{1}, η_{2}^{s} (u, μ) 〉 = 0 . \end{matrix}$ (2.28)

This finally yields $\begin{matrix} 〈 ζ_{1}^{1}, - (I - P_{1}^{-} (μ, 0)) v_{1, i}^{-} (u, ω, μ) (0) 〉 = o (e^{- 2 λ^{u} ω_{2, i}}) . \end{matrix}$

These considerations verify the $ζ_{1}^{1}$ -component of the representation of $ξ_{1, i} (u, ω, μ)$ . The $ζ_{1}^{2}$ -component of $ξ_{1, i} (u, ω, μ)$ follows by applying similar arguments to the terms containing $ζ_{1}^{2}$ . In doing so we find in particular(2.29) $\begin{matrix} c_{1}^{2} = - 〈 {\hat{η}}_{2}^{u} (μ), η_{2}^{s} (u_{i}, μ) 〉 \neq 0, \end{matrix}$ (2.29)

where $η_{2}^{s} (u_{i}, μ)$ is defined by (2.26). The vector ${\hat{η}}_{2}^{u}$ on the other hand is defined by(2.30) $\begin{matrix} Ψ_{1}^{-} (μ, t, 0) {(I - P_{1}^{-} (μ, 0))}^{*} ζ_{1}^{1} = e^{λ^{u} t} {\hat{η}}_{2}^{u} (μ) + o (e^{λ^{u} t}), as t \to - \infty . \end{matrix}$ (2.30)

Finally, the equivalent to (2.28) reads: $\begin{matrix} 〈 Ψ_{1}^{+} (μ, - ω_{1, i + 1}, 0) {(I - P_{1}^{+} (μ, 0))}^{*} ζ_{1}^{2}, η_{1}^{u} (u, μ) 〉 = 0 . \end{matrix}$

Next we consider the jump $ξ_{2, i}$ , see again (2.18): $\begin{matrix} ξ_{2, i} (u, ω, μ) = 〈 ζ_{2}, (I - P_{2}^{+} (u_{i}, μ, 0)) v_{2, i}^{+} (u, ω, μ) (0) - (I - P_{2}^{-} (u_{i}, μ, 0)) v_{2, i}^{-} (u, ω, μ) (0) 〉 ζ_{2} . \end{matrix}$

Considerations such as the ones for $ξ_{1, i}$ provide:(2.31) $\begin{matrix} c_{21} (u_{i}, μ) = 〈 {\hat{η}}_{1}^{u} (u_{i}, μ), η_{1}^{s} (μ) 〉 \neq 0 and c_{22} (u_{i}, μ) = 〈 {\hat{η}}_{2}^{s} (u_{i}, μ), η_{2}^{u} (μ) 〉 \neq 0 . \end{matrix}$ (2.31)

The quantities $η_{1}^{s} (μ)$ and $η_{2}^{u} (μ)$ are related to $q_{1}^{+} (μ)$ and $q_{1}^{-} (μ)$ in a way as given in (2.26) and (2.22), respectively. Alike the quantities ${\hat{η}}_{1}^{u} (u_{i}, μ)$ and ${\hat{η}}_{2}^{s} (u_{i}, μ)$ are related to the flows $Ψ_{2}^{-} (u_{i}, μ, t, 0)$ and $Ψ_{2}^{+} (u_{i}, μ, t, 0)$ applied to $ζ_{2}$ , cf. (2.30) and (2.23), respectively, for the corresponding expressions regarding $ξ_{1, i}$ . $□$

Remark 2.20:

We want to note that $ζ_{1}^{1}$ is located in the strong stable subspace of $Ψ_{1}^{-} (μ, t, 0)$ , as $t \to - \infty$ . A corresponding adaption of [13, Lemma 3.4] then allows an even better estimate as the one given in (2.27).

We remark that $ξ_{j, i}$ depends smoothly on all variables, cf. [16,28]. More precisely, as counterpart to [18, Lemma 4.6] or [13, Lemma 3.8] we have

Lemma 2.21:

The derivatives $D_{k} ξ_{1, i} (u, ω, μ)$ , $k = 1, 2, 3$ , of the jumps $ξ_{j, i}$ have the following form $\begin{matrix} D_{k} ξ_{1, i} (u, ω, μ) & = (D_{k} (c_{1}^{1} (u_{i + 1}, μ) e^{- 2 λ^{u} ω_{1, i + 1}}) + {\tilde{R}}_{1, i}^{1} (u, ω, μ)) ζ_{1}^{1} \\ + (D_{k} (- c_{1}^{2} (u_{i}, μ) e^{- 2 λ^{u} ω_{2, i}}) + {\tilde{R}}_{1, i}^{2} (u, ω, μ)) ζ_{1}^{2}, \\ D_{k} ξ_{2, i} (u, ω, μ) & = (D_{k} (c_{21} (u_{i}, μ) e^{2 λ^{s} ω_{1, i}} - c_{22} (u_{i}, μ) e^{2 λ^{s} ω_{2, i}}) + {\tilde{R}}_{2, i} (u, ω, μ)) ζ_{2}, \end{matrix}$

where $\begin{matrix} {\tilde{R}}_{1, i}^{1} (u, ω, μ) & = o (e^{- 2 λ^{u} ω_{1, i + 1}}) + o (e^{- 2 λ^{u} ω_{2, i}}) \\ {\tilde{R}}_{1, i}^{2} (u, ω, μ) & = o (e^{- 2 λ^{u} ω_{1, i + 1}}) + o (e^{- 2 λ^{u} ω_{2, i}}) \\ {\tilde{R}}_{2, i} (u, ω, μ) & = o (e^{2 λ^{s} ω_{1, i}}) + o (e^{2 λ^{s} ω_{2, i}}) . \end{matrix}$

Note that here the derivatives with respect to $u$ are also considered. Statements concerning this matter are due to considerations in [16].

Finally, we consider the relations between the coefficients $c_{1}^{1}$ and $c_{1}^{2}$ or $c_{21}$ and $c_{22}$ , respectively, which are due to the reversing symmetry.

Lemma 2.22:

Assume (H1)–(H12). Then

(i)	$c_{1}^{1} (u, μ) = c_{1}^{2} (u, μ) = : a (u, μ)$ ,
(ii)	$c_{21} (u, μ) = - c_{22} (u, μ) = : c (u, μ)$ .

ProofThe definitions $c_{1}^{1}$ and $c_{1}^{2}$ are given in (2.24) and (2.29), respectively. Due to the reversing symmetry of the vector field, cf. (H1), Lemmas 2.7 and 2.8 we have $R η_{1}^{u} (u, μ) = η_{2}^{s} (u, μ)$ . Similarly, taking into consideration $ζ_{1}^{2} = R ζ_{1}^{1}$ , we find $R {\hat{η}}_{1}^{s} (μ) = {\hat{η}}_{2}^{u} (μ)$ . The statement (i) now follows from the R-invariance of the scalar product, cf. Section 2.1.

The statement (ii) follows by the same type of arguments but this time applied to the quantities appearing in (2.31).

Summarising the results of this section we find the following. Assume (H1)–(H12), then the determination Equation (2.15) for actual orbits staying for all time close to the original T-point is equivalent to(2.32) $\begin{matrix} 0 & = μ_{1} + a (u_{i + 1}, μ) e^{- 2 λ^{u} ω_{1, i + 1}} + R_{1, i}^{1} (u, ω, μ) \\ 0 & = - μ_{1} - a (u_{i}, μ) e^{- 2 λ^{u} ω_{2, i}} + R_{1, i}^{2} (u, ω, μ) \\ 0 & = μ_{2} - u_{i}^{2} + c (u_{i}, μ) (e^{2 λ^{s} ω_{1, i}} + e^{2 λ^{s} ω_{2, i}}) + R_{2, i} (u, ω, μ), \\ i & \in Z . \end{matrix}$ (2.32)

We remark that the first two equations come from setting the jump $Ξ_{1, i}$ in $Σ_{1}$ to zero while the third equation comes from setting the jump $Ξ_{2, i}$ in $Σ_{2}$ to zero.

3. Dynamics near $Γ$

Throughout this section we assume Hypotheses (H1)–(H12). That is, we assume in particular that $Γ$ is non-elementary and that $λ^{u}$ is real and is the leading eigenvalue.

3.1. Shift dynamics – Proof of Theorem 1.1

Let $u$ and $ω$ be sequences according to Theorem 2.9. We define(3.1) $\begin{matrix} r_{j, i} : = e^{- 2 λ^{u} ω_{j, i}}, j = 1, 2, i \in Z, r : = {((r_{1, i}, r_{2, i}))}_{i \in Z} . \end{matrix}$ (3.1)

With that (2.32) reads(3.2) $\begin{matrix} 0 & = μ_{1} + a (u_{i + 1}, μ) r_{1, i + 1} + {\hat{R}}_{1, i}^{1} (u, r, μ) \\ 0 & = - μ_{1} - a (u_{i}, μ) r_{2, i} + {\hat{R}}_{1, i}^{2} (u, r, μ) \\ 0 & = μ_{2} - u_{i}^{2} + c (u_{i}, μ) (r_{1, i}^{δ^{s}} + r_{2, i}^{δ^{s}}) + {\hat{R}}_{2, i} (u, r, μ), \\ i & \in Z . \end{matrix}$ (3.2)

The residual terms ${\hat{R}}_{1, i}^{j}$ and ${\hat{R}}_{2, i}$ arise from the corresponding $R_{1, i}^{j}$ and $R_{2, i}$ by applying (3.1). Furthermore we assume the following sign condition for the coefficients a and c:(3.3) $\begin{matrix} a (0, 0) < 0, c (0, 0) < 0 . \end{matrix}$ (3.3)

This condition merely determines in which quadrant of the $(μ_{1}, μ_{2})$ -plane shift dynamics exists.

Assuming (3.3) we solve, for positive $μ_{1}$ , the subsystem of (3.2) consisting of the first two equations (for each $i \in Z$ ), for $\begin{matrix} r = r (u, μ), where r_{j, i} (u, μ) = - \frac{1}{a (0, 0)} μ_{1} + o (μ_{1}), j = 1, 2, i \in Z . \end{matrix}$

It remains to solve the remaining third equations of (3.2), in which we plug in the solving function $r (u, μ)$ :(3.4) $\begin{matrix} 0 = μ_{2} - u_{i}^{2} + c (u_{i}, μ) (r_{1, i} (u, μ))^{δ^{s}} + c (u_{i}, μ) (r_{2, i} (u, μ))^{δ^{s}} + {\hat{R}}_{2, i} (u, r (u, μ), μ), i \in Z . \end{matrix}$ (3.4)

Equation (3.4) can be written as(3.5) $\begin{matrix} 0 = μ_{2} - u_{i}^{2} - \frac{2 c (u_{i}, μ)}{a (0, 0)} μ_{1}^{δ^{s}} + O (μ_{1}^{δ δ^{s}}), δ > 1, i \in Z . \end{matrix}$ (3.5)

Consider the truncated equation $\begin{matrix} 0 = ξ_{s h} (u, μ), ξ_{s h} (u, μ) : = μ_{2} - u^{2} - \frac{2 c (u, μ)}{a (0, 0)} μ_{1}^{δ^{s}} . \end{matrix}$

This equation can be solved for $μ_{2}$ :(3.6) $\begin{matrix} μ_{2} = u^{2} + {\hat{C}}_{s h} (u, μ_{1}), \end{matrix}$ (3.6)

with ${\hat{C}}_{s h} (u, μ_{1}) = O (μ_{1}^{δ^{s}})$ and $\frac{\partial}{\partial u} {\hat{C}}_{s h} (u, μ_{1}) = O (μ_{1}^{δ^{s}})$ . Hence the right-hand side of (3.6) has a unique minimum $u_{s h}^{*} (μ_{1})$ . With that we define $\begin{matrix} {μ_{2}}_{s h}^{*} (μ_{1}) : = (u_{s h}^{*} (μ_{1}))^{2} + {\hat{C}}_{s h} (u_{s h}^{*} (μ_{1}), μ_{1}) = C_{s h} μ_{1}^{δ^{s}} + o (μ_{1}^{δ^{s}}), \end{matrix}$

where $C_{s h}$ is a constant which is different from zero. Now, fix some(3.7) $\begin{matrix} μ = (μ_{1}, μ_{2}) : μ_{1} > 0 and μ_{2} > {μ_{2}}_{s h}^{*} (μ_{1}), \end{matrix}$ (3.7)

and let $u^{+} (μ)$ , $u^{-} (μ)$ be the two solutions of (3.6), or in other words $\begin{matrix} ξ_{s h} (u, μ) = 0 \Leftrightarrow u \in {u^{+} (μ), u^{-} (μ)} . \end{matrix}$

We find(3.8) $\begin{matrix} u^{\pm} (μ) = \pm \sqrt{μ_{2} - E μ_{1}^{δ^{s}}} + O (μ_{1}^{δ^{s}}), \end{matrix}$ (3.8)

where $E > 0$ is some constant. To verify (3.8) just write $u = \pm \sqrt{μ_{2} - E μ_{1}^{δ^{s}}} + v$ and solve equation $ξ_{s h} (\pm \sqrt{μ_{2} - E μ_{1}^{δ^{s}}} + v, μ) = 0$ for v.

Furthermore, according to the definition of $ξ_{s h}$ we find(3.9) $\begin{matrix} D_{1} ξ_{s h} (u^{\pm}, μ) = - 2 (u^{\pm} + \frac{D_{1} c (u^{\pm}, μ)}{a (0, 0)} μ_{1}^{δ^{s}}) . \end{matrix}$ (3.9)

With that we define sequences(3.10) $\begin{matrix} u^{*} : = (u_{i}^{*})_{i \in Z}, where u_{i}^{*} \in {u^{+} (μ), u^{-} (μ)} . \end{matrix}$ (3.10)

Our goal is to show that there are $μ_{2}$ (for fixed given $μ_{1}$ ) and $ϵ$ such that (3.4) can be uniquely solved in a closed ball $B (u^{*}, ϵ) \subset l_{U}^{\infty}$ centred at $u^{*}$ with radius(3.11) $\begin{matrix} ϵ < \frac{1}{2} | u^{+} (μ) - u^{-} (μ) | . \end{matrix}$ (3.11)

By $l_{U}^{\infty}$ we denote the set of $l^{\infty}$ sequences in U equipped with the supremum norm.

Lemma 3.1:

There is a number $N_{s d} \in N$ such that for $μ = (μ_{1}, μ_{2})$ , $μ_{1} > 0$ and $μ_{2} > N_{s d} μ_{1}^{δ^{s}}$ we have the following: there is an $ϵ < \frac{1}{2} | u^{+} (μ) - u^{-} (μ) |$ such that for each sequence $u^{*}$ according to (3.10) there is a unique solution $(u_{u^{*}}, r_{u^{*}})$ of (3.2) with $u_{u^{*}} \in B (u^{*}, ϵ)$ .

By (3.1) the sequence $r_{u^{*}}$ defines a unique sequence $ω_{u^{*}}$ . Note that both $u_{u^{*}}$ and $ω_{u^{*}}$ depend on $μ$ , although we omit this dependence in our notation.

In other words the lemma says that (for the corresponding $μ$ and $u^{*}$ ) there is a unique orbit $\begin{matrix} X (u^{*}, μ) : = X (u_{u^{*}}, ω_{u^{*}}, μ) \end{matrix}$

intersecting $Σ_{2}$ near $u^{+} (μ)$ or $u^{-} (μ)$ , respectively, in the order prescribed by $u^{*}$ . Note that $X (u_{u^{*}}, ω_{u^{*}}, μ)$ denotes a Lin orbit which is in this case a real orbit. We also define a corresponding solution $x (u^{*}, μ) (\cdot)$ of (1.1) by(3.12) $\begin{matrix} x (u^{*}, μ) (0) = x_{2, 0}^{+} (u_{u^{*}}, ω_{u^{*}}, μ) (0) = x_{2, 0}^{-} (u_{u^{*}}, ω_{u^{*}}, μ) (0) \in Σ_{2} . \end{matrix}$ (3.12)

Proof of Lemma 3.1:

We employ the Banach fixed point theorem to solve (3.5).

Using Taylor expansion w.r.t. u $\begin{matrix} ξ_{s h} (u_{i}, μ) = D_{1} ξ_{s h} (u_{i}^{*}, μ) (u_{i} - u_{i}^{*}) + O ({(u_{i} - u_{i}^{*})}^{2}) \end{matrix}$

we rewrite Equation (3.5) as follows(3.13) $\begin{matrix} u_{i} = u_{i}^{*} + (D_{1} ξ_{s h} (u_{i}^{*}))^{- 1} (O ({(u_{i} - u_{i}^{*})}^{2}) + O (μ_{1}^{δ δ^{s}})) = : T_{i} (u), i \in Z . \end{matrix}$ (3.13)

Note that the $O (μ_{1}^{δ δ^{s}})$ -term appearing in (3.13) also depends on $u$ and $μ_{2}$ .

In what follows we show that (3.13) can be read as a fixed point equation $\begin{matrix} u = T (u) = {(T_{i} (u))}_{i \in Z}, T (\cdot) : B (u^{*}, ϵ) \subset l_{U}^{\infty} \to B (u^{*}, ϵ) \end{matrix}$

which satisfies the assumptions of the Banach fixed point theorem.

First we show that there are $μ_{2}$ (for fixed given $μ_{1}$ ) and $ϵ$ such that $T$ maps $B (u^{*}, ϵ)$ into itself: according to (3.13) we have $\begin{matrix} | T_{i} (u) - u_{i}^{*} | \leq |(D_{1} ξ_{s h} (u_{i}^{*}))^{- 1}| (|O ({(u_{i} - u_{i}^{*})}^{2})| + |O (μ_{1}^{δ δ^{s}})|) \end{matrix}$

Now we write $μ_{2} = m E μ_{1}^{δ^{s}}$ , $m > 1$ , where m has to be large enough such that (3.7) is still satisfied. Taking into consideration (3.8) and (3.9), we find $\begin{matrix} u^{\pm} (μ) = \pm \sqrt{(m - 1) E} μ_{1}^{\frac{δ^{s}}{2}} + O (μ_{1}^{δ^{s}}), \end{matrix}$

and $\begin{matrix} D_{1} ξ_{s h} (u^{\pm}, μ) = \mp 2 \sqrt{(m - 1) E} μ_{1}^{\frac{δ^{s}}{2}} + O (μ_{1}^{δ^{s}}) . \end{matrix}$

Hence, for $κ \in (1 / 2, 1)$ and sufficiently small $μ_{1}$ (3.14) $\begin{matrix} | u_{i} - u_{i}^{*} | \leq M μ_{1}^{- κ δ^{s}} (|O ({(u_{i} - u_{i}^{*})}^{2})| + |O (μ_{1}^{δ δ^{s}})|), i \in Z . \end{matrix}$ (3.14)

Now, choose $\begin{matrix} ϵ : = μ_{1}^{λ δ^{s}}, 1 / 2 < κ < λ, κ + λ < δ, \end{matrix}$

and choose $μ_{1}$ small enough such that both (3.11) and $\begin{matrix} ϵ^{- κ / λ} (|O (ϵ^{2})| + |O (ϵ^{δ / λ})|) < ϵ \end{matrix}$

are satisfied. Then it follows from (3.14) that for those $μ_{1}$ the operator $T$ maps $B (u^{*}, ϵ)$ into itself.

It remains to show that $T$ is contractive. This can be done by examining $‖ D T (u) ‖$ . We first note that $T$ is indeed differentiable (as mapping $l_{U}^{\infty} \to l_{U}^{\infty}$ ). This follows from corresponding differentiability of $Ξ = {(Ξ_{1, i} (u, ω, μ), Ξ_{2, i} (u, ω, μ))}_{i \in Z}$ , which we get from considerations in [16], cf. also Lemma 2.21.

Now, inspecting the right-hand side of (3.13), and if necessary decreasing $ϵ$ (and so decreasing $μ_{1}$ ) further, we find that $T$ is contractive on $B (u^{*}, ϵ)$ .

Note, that by our construction $N_{s d}$ can be defined by $N_{s d} : = m E$ . Note further that for sufficiently small $μ_{1}$ , the constant m can be chosen independently of $μ_{1}$ . $□$

3.1.1. Symbolic dynamics

Let $N_{s d}$ , $μ$ and $u^{*}$ be in accordance with Lemma 3.1. We define a shift operator $σ$ on $S_{μ}^{2} : = {u^{+} (μ), u^{-} (μ)}^{Z}$ , equipped with the product topology, by(3.15) $\begin{matrix} \begin{matrix} σ : & {u^{+} (μ), u^{-} (μ)}^{Z} & \to {u^{+} (μ), u^{-} (μ)}^{Z} \\ u^{*} = {(u_{i}^{*})}_{i \in Z} & \mapsto v^{*} = {(v_{i}^{*})}_{i \in Z}, & v_{i}^{*} = u_{i + 1}^{*} . \end{matrix} \end{matrix}$ (3.15)

Let $u^{*} = {(u_{i} *)}_{i \in Z} \in {u^{+} (μ), u^{-} (μ)}^{Z}$ . Then the $σ$ -orbit $O_{σ} (u^{*})$ through $u^{*}$ is defined by $\begin{matrix} O_{σ} (u^{*}) = \{σ^{n} (u^{*}) : n \in Z\} . \end{matrix}$

Hence, $O_{σ} (u^{*})$ is N-periodic, or in other words, $u^{*}$ is an N-periodic point if $u_{i + N}^{*} = u_{i}^{*}$ , for all $i \in Z$ . That means that the sequence ${(u_{i} *)}_{i \in Z}$ is N-periodic. In this case we write $\begin{matrix} {(u_{i}^{*})}_{i \in Z} = (\bar{u_{1}^{*} \dots u_{N}^{*}}) . \end{matrix}$

Furthermore we define the following subset of $Σ_{2}$ $\begin{matrix} S_{μ} : = \{x (u^{*}, μ) (0) : u^{*} \in {u^{+} (μ), u^{-} (μ)}^{Z}\} . \end{matrix}$

On $S_{μ}$ we define a first return map $Π_{μ}$ , see also Figure 5, $\begin{matrix} \begin{matrix} Π_{μ} : & S_{μ} & \to S_{μ} \\ x (u^{*}, μ) (0) & \mapsto x (u^{*}, μ) (2 (ω_{2, 0} + ω_{1, 1})) . \end{matrix} \end{matrix}$

Finally we define a one-to-one mapping ${u^{+} (μ), u^{-} (μ)}^{Z} \to S_{μ}$ (see also (3.12))(3.16) $\begin{matrix} \begin{matrix} h_{μ} : & {u^{+} (μ), u^{-} (μ)}^{Z} & \to S_{μ} \\ u^{*} & \mapsto x (u^{*}, μ) (0) . \end{matrix} \end{matrix}$ (3.16)

Lemma 3.2:

The systems $(S_{μ}, Π_{μ})$ and $({u^{+} (μ), u^{-} (μ)}^{Z}, σ)$ are topologically conjugated.

ProofWe show that the mapping $h_{μ}$ introduced in (3.16) is a topological conjugation. To this end we first show that $h_{μ}$ is a conjugation:(3.17) $\begin{matrix} Π_{μ} \circ h_{μ} = h_{μ} \circ σ . \end{matrix}$ (3.17)

Equation (3.17) means that for all $u^{*}$ $\begin{matrix} Π_{μ} (x (u^{*}, μ) (0)) = x (σ u^{*}, μ) (0) . \end{matrix}$

The point $x (u^{*}, μ) (0)$ is the starting point of the orbit $X (u_{u^{*}}, ω_{u^{*}}, μ)$ , while $x (σ u^{*}, μ) (0)$ is the starting point of the orbit $X (u_{σ u^{*}}, ω_{σ u^{*}}, μ)$ , cf. (3.12). Furthermore, $Π_{μ} (x (u^{*}, μ) (0)) = x (u^{*}, μ) (2 (ω_{2, 0} + ω_{1, 1}))$ is the starting point of the orbit $X (σ u_{u^{*}}, σ ω_{u^{*}}, μ)$ . So, to verify (3.17) it remains to make clear that $\begin{matrix} X (σ u_{u^{*}}, σ ω_{u^{*}}, μ) = X (u_{σ u^{*}}, ω_{σ u^{*}}, μ) . \end{matrix}$

This follows from the uniqueness statement in Lemma 3.1 which says that there is a one-to-one correspondence of $u^{*}$ to sequences $(u_{u^{*}}, ω_{u^{*}})$ . Clearly $σ u^{*}$ is related to $(σ u_{u^{*}}, σ ω_{u^{*}})$ . Hence, due to the uniqueness $(σ u_{u^{*}}, σ ω_{u^{*}}) = (u_{σ u^{*}}, ω_{σ u^{*}})$ .

To complete the proof of the lemma it remains to show that $h_{μ}$ is a homeomorphism. This part of the proof runs along the lines of [13, Section 4.2]. Here we confine ourselves to explain the major points.

Since $({u^{+} (μ), u^{-} (μ)}^{Z}$ is compact (with respect to the product topology) and $S_{μ}$ is Hausdorff it is enough to show that $h_{μ}$ is continuous, cf. [8, Chap. XI, Theorem 2.1]. To show the continuity of $h_{μ}$ we proceed as in the proof of [13, Lemma 4.7]. However, there is an extra difficulty due to the internal parameter u. To deal with that we proceed as in [17, Section 3], where a similar problem in the context of discrete systems has been considered. Now, let $u^{*}, w^{*} \in {u^{+} (μ), u^{-} (μ)}^{Z}$ , and let $u_{u^{*}} = {(u_{i, u^{*}})}_{i \in Z}$ . Then, according to (3.16), (3.12) and (2.17) we find(3.18) $\begin{matrix} ‖ h_{μ} (u^{*}) - h_{μ} (w^{*}) ‖ \leq & ‖ q_{2}^{+} (u_{0, u^{*}}, μ) (0) - q_{2}^{+} (u_{0, w^{*}}, μ) (0) ‖ \\ + ‖ v_{2, 0}^{+} (u_{u^{*}}, ω_{u^{*}}, μ) (0) - v_{2, 0}^{+} (u_{w^{*}}, ω_{w^{*}}, μ) (0) ‖ . \end{matrix}$ (3.18)

These terms can be estimated similarly as the related terms in the proof of [17, Lemma 3.3]. We refer in particular to equation (20) in that proof. From that we infer that both addends on the right-hand side of (3.18) are of order O(1 / k) if $u^{*}$ and $w^{*}$ coincide on a block of length $k^{2}$ centred at $i = 0$ . Therefore $h_{μ}$ is continuous.

3.1.2. Reversible symbolic dynamics

In what follows we identify the symbol $u^{+} (μ)$ with $+$ and similarly $u^{-} (μ)$ with $-$ . We similarly also identify $S_{μ}^{2} : = {u^{+} (μ), u^{-} (μ)}^{Z}$ with $S^{2} : = {-, +}^{Z}$ . In this way we may consider the conjugation $h_{μ}$ , cf. (3.16), as being defined on $S^{2}$ , and we may consider sequences $u^{*}$ as elements of $S^{2}$ .

Let $S^{2} : = {-, +}^{Z}$ be equipped with the product topology. Consider the system $(S^{2}, σ)$ , where $σ$ is the shift operator defined as in (3.15). The mapping $\begin{matrix} \begin{matrix} R : & S^{2} & \to S^{2} \\ s & \mapsto R s, & {(R s)}_{i} = s_{- i} \end{matrix} \end{matrix}$

is an involution. We have $\begin{matrix} Fix R = {s \in S^{2} : s_{i} = s_{- i}, i \in Z} . \end{matrix}$

Lemma 3.3:

The system $(S^{2}, σ)$ is reversible w.r.t. $R$ , i.e. $σ \circ R = R \circ σ^{- 1}$ .

Proof $σ \circ R ({(s_{i})}_{i \in Z}) = {(s_{- i + 1})}_{i \in Z} = R \circ σ^{- 1} ({(s_{i})}_{i \in Z})$ .

For $s \in S^{2}$ the $σ$ -orbit $O_{σ} (s)$ is $R$ -symmetric if $R (O_{σ} (s)) = O_{σ} (s)$ . So,(3.19) $\begin{matrix} O_{σ} (s) is symmetric \Leftrightarrow \exists N \in Z : σ^{N} R s = s . \end{matrix}$ (3.19)

Lemma 3.4:

The $σ$ -orbit $O_{σ} (s)$ is $R$ -symmetric if and only if(3.20) $\begin{matrix} (a) \exists \hat{s} \in O_{σ} (s) : R \hat{s} = \hat{s} or (b) \exists \hat{s} \in O_{σ} (s) : R \hat{s} = σ \hat{s} \end{matrix}$ (3.20)

ProofAccording to (3.19) it remains to show that the symmetry of the orbit implies (3.20). Let $O_{σ} (s)$ be $R$ -symmetric and let $N \in Z$ such that $R s = σ^{- N} s$ , cf. (3.19). Therefore, for $k \in Z$ we have $R σ^{- k} s = σ^{- N + k} s$ .

Now, let |N| be even. For $k = N / 2$ we find $R σ^{- N / 2} s = σ^{- N / 2} s$ . If on the contrary |N| is odd we choose $k = (N + 1) / 2$ and find $R σ^{- (N + 1) / 2} s = σ σ^{- (N + 1) / 2} s$ .

Condition (a) in (3.20) says that the symmetric orbit $O_{σ} (s)$ intersects $Fix R$ in $\hat{s}$ . Note that the sequence $\hat{s}$ is symmetric with respect to the reflection element $s_{0}$ . By contrast, the sequence $\hat{s}$ in condition (b) in (3.20) does not belong to $Fix R$ . This sequence is symmetric with respect to the ‘gap between $s_{0}$ and $s_{1}$ ’.

Lemma 3.5:

If $O_{σ} (s)$ is a $R$ -symmetric aperiodic orbit then exactly one of the conditions stated in (3.20) is satisfied for exactly one $\hat{s}$ .

ProofWe show, provided both conditions in (3.20) are satisfied or one these conditions is satisfied for two different ${\hat{s}}^{1}$ , ${\hat{s}}^{2}$ then necessarily $O_{σ} (s)$ is periodic.

Assume that there are ${\hat{s}}^{1}$ , ${\hat{s}}^{2}$ such that $R {\hat{s}}^{i} = {\hat{s}}^{i}$ , $i = 1, 2$ , and that ${\hat{s}}^{2} = σ^{N} {\hat{s}}^{1}$ . Under these assumptions we find $\begin{matrix} σ (σ^{N} {\hat{s}}^{1}) = σ {\hat{s}}^{2} = σ R {\hat{s}}^{2} = R σ^{- 1} {\hat{s}}^{2} = R σ^{N - 1} {\hat{s}}^{1} = σ^{- N + 1} R {\hat{s}}^{1} = σ^{- N + 1} {\hat{s}}^{1} . \end{matrix}$

Next assume that there are ${\hat{s}}^{1}$ , ${\hat{s}}^{2}$ such that $R {\hat{s}}^{i} = σ {\hat{s}}^{i}$ , $i = 1, 2$ , and that ${\hat{s}}^{2} = σ^{N} {\hat{s}}^{1}$ . Under these assumptions we find $\begin{matrix} σ (σ^{N} {\hat{s}}^{1}) = σ {\hat{s}}^{2} = R {\hat{s}}^{2} = R σ^{N} {\hat{s}}^{1} = σ^{- N} R {\hat{s}}^{1} = σ^{- N} σ {\hat{s}}^{1} = σ^{- N + 1} {\hat{s}}^{1} . \end{matrix}$

Assume finally that there are ${\hat{s}}^{1}$ such that $R {\hat{s}}^{1} = {\hat{s}}^{1}$ and ${\hat{s}}^{2}$ such that $R {\hat{s}}^{2} = σ {\hat{s}}^{2}$ , and that ${\hat{s}}^{2} = σ^{N} {\hat{s}}^{1}$ . Under these assumptions we find $\begin{matrix} σ (σ^{N} {\hat{s}}^{1}) = σ {\hat{s}}^{2} = R {\hat{s}}^{2} = R σ^{N} {\hat{s}}^{1} = σ^{- N} R {\hat{s}}^{1} = σ^{- N} {\hat{s}}^{1} . \end{matrix}$

The proof of Lemma 3.5 gives a characterisation of symmetric periodic orbits.

Corollary 3.6:

The orbit $O_{σ} (s)$ is a symmetric periodic orbit if and only if either both conditions in (3.20) are satisfied or one of these conditions is satisfied for two different ${\hat{s}}^{1}$ , ${\hat{s}}^{2}$ .

ProofBy the proof of Lemma 3.5 it remains to show that the symmetry of a periodic orbit implies that either both conditions in (3.20) are satisfied or one these conditions is satisfied for two different ${\hat{s}}^{1}$ , ${\hat{s}}^{2}$ .

Let $O_{σ} (s)$ be N-periodic and symmetric.

First assume that there exists a $\hat{s} \in O_{σ} (s)$ such that $R \hat{s} = \hat{s}$ , cf. condition (a) in (3.20). If $N = 2 K + 1$ , we find that due to the symmetry we have $σ^{- K} \hat{s} = R σ^{K} \hat{s}$ . Furthermore, since $O_{σ} (s)$ is $(2 K + 1)$ -periodic we have $σ (σ^{K} \hat{s}) = σ^{- K} \hat{s}$ . Hence condition (b) in (3.20) is satisfied with $σ^{K} \hat{s}$ .

If $N = 2 K + 2$ , we find that $\begin{matrix} O_{σ} (s) = {σ^{- K} \hat{s}, \dots, σ^{- 1} \hat{s}, \hat{s}, σ \hat{s}, \dots, σ^{K} \hat{s}, σ^{K + 1} \hat{s}} . \end{matrix}$

Due to the condition $R \hat{s} = \hat{s}$ we find that condition (a) in (3.20) is satisfied with $σ^{K + 1} \hat{s}$ .

Next assume that there exists a $\hat{s} \in O_{σ} (s)$ such that $R \hat{s} = σ \hat{s}$ , cf. condition (b) in (3.20). If $N = 2 K + 2$ , we find that $\begin{matrix} O_{σ} (s) = {σ^{- K} \hat{s}, \dots, σ^{- 1} \hat{s}, \hat{s}, σ \hat{s}, \dots, σ^{K} \hat{s}, σ^{K + 1} \hat{s}} . \end{matrix}$

Due to the condition $R \hat{s} = σ \hat{s}$ we find that condition (b) in (3.20) is satisfied with $σ^{K + 1} \hat{s}$ .

If $N = 2 K + 1$ , we find that $\begin{matrix} O_{σ} (s) = {σ^{- K + 1} \hat{s}, \dots, σ^{- 1} \hat{s}, \hat{s}, σ \hat{s}, \dots, σ^{K} \hat{s}, σ^{K + 1} \hat{s}} . \end{matrix}$

Due to the condition $R \hat{s} = σ \hat{s}$ we find that condition (a) in (3.20) is satisfied with $σ^{K + 1} \hat{s}$ .

Indeed there are symmetric periodic orbits which intersect $Fix R$ only once (both conditions in (3.20) are satisfied), and there are even symmetric periodic orbits which do not intersect $Fix R$ at all. Examples for those orbits are $O_{σ} ((\bar{+ + -}))$ or $O_{σ} ((\bar{+ + - -}))$ , respectively.

Let $S_{R}^{2}$ denote the set of all sequences in $S^{2}$ whose $σ$ -orbit is symmetric $\begin{matrix} S_{R}^{2} : = {s \in S^{2} : O_{σ} (s) is symmetric} . \end{matrix}$

Remark 3.7:

The set $S_{R}^{2}$ is a nonempty proper subset of $S^{2}$ : With the examples given above it is clear that $S_{R}^{2}$ is nonempty. In fact all periodic orbits up to period five are contained in $S_{R}^{2}$ . In order to verify that there are sequences $s$ in $S^{2}$ whose $σ$ -orbit is asymmetric consider for instance the 7-periodic sequence $s = (\bar{+ + + - - + -})$ .

By construction we find that $S_{R}^{2}$ is $σ$ -invariant. In [16, Section 6.4.4] it has been shown that, although $S_{R}^{2}$ is not closed, the system $(S_{R}^{2}, σ)$ exhibits chaotic dynamics in the sense of Devaney. First we recall Devaney’s definition of a chaotic system, cf. [7]: Let X be a metric space and f be a homeomorphism on X. The discrete dynamical system (X, f) is chaotic if

(i)	f is topologically transitive;
(ii)	The periodic points of f are dense in X;
(iii)	f has sensitive dependence on initial conditions.

The result of Banks et al. [2], states that (i) and (ii) imply sensitive dependence on initial conditions. So it remains to verify that

(S_{R}^{2}, σ)

is topologically transitive and that the periodic points are dense.

Lemma 3.8:

[16, Lemma 6.4.17] Consider $(S_{R}^{2}, σ)$ . The set of periodic points is dense in $S_{R}^{2}$ .

ProofWe show that the set of those periodic points is even dense in $S^{2}$ . Let $s : = {(s_{i})}_{i \in Z} \in S^{2}$ . Then $(\bar{s_{n} \dots s_{0} \dots s_{- n} + s_{- n} \dots_{∙} s_{0} \dots s_{n}})$ is a symmetric periodic point coinciding with $s$ on segment of length 2n+1 around ‘ $_{∙}$ ’.

Lemma 3.9:

[16, Lemma 6.4.18] There is a dense orbit in $(S_{R}^{2}, σ)$ .

ProofThe proof is very similar to the proof that there is a dense orbit in $(S^{2}, σ)$ . Let $s^{k} : = {s_{i}^{k} = s_{i, 1}^{k}, \dots, s_{i, k}^{k} : s_{i, j}^{k} \in {-, +}, i \in {1, \dots, 2^{k}}}$ be the set of all finite segments of length k. Introduce an order in $s^{k}$ in the following way: $s_{i}^{k} < s_{j}^{l} \Leftrightarrow k < l or k = l, i < j .$

Finally let ${\bar{s}}_{i}^{k} = s_{i, k}^{k}, \dots, s_{i, 1}^{k}$ . The orbit through $s : = (\dots {\bar{s}}_{1}^{2} {\bar{s}}_{2}^{1} {\bar{s}}_{1}^{1}_{∙} + s_{1}^{1} s_{2}^{1} s_{1}^{2} \dots s_{4}^{2} \dots)$ is dense in $S_{R}^{2}$ .

An immediate consequence of this lemma is

Corollary 3.10:

[16, Corollary 6.4.19] The system $(S_{R}^{2}, σ)$ is topologically transitive.

Indeed, $S_{R}^{2}$ is not closed because the orbit constructed in Lemma 3.9 is even dense in $S^{2}$ . Moreover the addressed result of Banks et al. implies that systems which are topologically conjugated to $(S_{R}^{2}, σ)$ are chaotic as well.

Lemma 3.11:

For the map $h_{μ} : S^{2} \to S_{μ} \subset Σ_{2}$ we have $h_{μ} \circ R = R \circ h_{μ}$ .

ProofThe statement of the lemma immediately translates into $x (R u^{*}, μ) (0) = R (x (u^{*}, μ) (0))$ . This equality follows with Lemma 2.13 and the uniqueness statement in Lemma 3.1.

This yields the following:

Corollary 3.12:

The statement of Lemma 3.11 implies

(i)

$R (S_{μ}) = S_{μ}$ .

(ii)

The system $(S_{μ}, Π_{μ})$ is reversible w.r.t. to the involution R, i.e. $Π_{μ} \circ R = R \circ Π_{μ}^{- 1}$ .

(iii)

The $σ$ -orbit through $u^{*}$ is $R$ -symmetric if and only if the $Π_{μ}$ -orbit through $h_{μ} (u^{*})$ is R-symmetric. Then according to Lemma 3.4 we have the following possibilities:

	(a)	$R {\hat{u}}^{} = {\hat{u}}^{}$ $\Leftrightarrow$ $h_{μ} ({\hat{u}}^{}) = x ({\hat{u}}^{}, μ) (0) \in Fix R \cap Σ_{2}$ , for ${\hat{u}}^{} \in O_{σ} (u^{})$ ,
	(b)	$R {\hat{u}}^{} = σ {\hat{u}}^{}$ $\Leftrightarrow$ $x ({\hat{u}}^{}, μ) (2 ω_{2, 0}) \in Fix R \cap Σ_{1}$ , for ${\hat{u}}^{} \in O_{σ} (u^{*})$ .

Proof:

(i)

According to Lemma 3.11 we have $h_{μ} (R (S^{2})) = R (h_{μ} (S^{2}))$ . With $R (S^{2}) = S^{2}$ and $h_{μ} (S^{2}) = S_{μ}$ we prove the statement.

(ii)

According to Lemma 3.3 we have $σ R = R σ^{- 1}$ , where we substitute $σ = h_{μ}^{- 1} Π_{μ} h_{μ}$ , cf. (3.17). Taking the statement of Lemma 3.11 into consideration we have $h_{μ}^{- 1} Π_{μ} R h_{μ} = h_{μ}^{- 1} R Π_{μ}^{- 1} h_{μ}$ . This gives the statement (ii).

(iii)

With (3.17) and Lemma 3.11 we find $h_{μ} σ^{N} R (u^{*}) = Π_{μ}^{N} R h_{μ} (u^{*})$ . Now, the statement follows with (3.19) and the analogous statement for $Π_{μ}$ -orbits.

	(a)	Let ${\hat{u}}^{} \in O_{σ} (u^{}) \cap Fix R$ . Of course, cf. Lemma 3.2, $h_{μ} ({\hat{u}}^{}) \in O_{Π_{μ}} (h_{μ} (u^{}))$ . According to Lemma 3.11 we find $R h_{μ} ({\hat{u}}^{}) = h_{μ} (R {\hat{u}}^{}) = h_{μ} ({\hat{u}}^{*})$ .
	(b)	By the conjugacy $h_{μ}$ it is clear that the $u^{}$ -symmetry implies the R-symmetry of the corresponding $Π_{μ}$ -orbit and hence also of the corresponding f-orbit. Due to the symmetry property $R {\hat{u}}^{} = σ {\hat{u}}^{*}$ we invoke Lemma 2.16 with $i_{0} = - 1$ to find that $ω_{j, i} = ω_{j + 1, - i + 1}$ . Finally this implies the statement. $□$

Remark 3.13:

Recall that $Π_{μ}$ -orbits correspond to f-orbits, i.e. to orbits of (1.1) and that symmetric periodic f-orbits intersect $Fix R$ exactly twice and a symmetric aperiodic f-orbit intersects $Fix R$ exactly once. Those f-orbits may intersect $Fix R$ within $Σ_{1}$ or $Σ_{2}$ . Which case is at hand can be read from ‘the symmetry of $u^{*}$ ’ according to (iii)(a) or (iii)(b) in the foregoing corollary. Case (iii)(a) implies an intersection of the corresponding f-orbit with $Fix R \cap Σ_{2}$ , while case (iii)(b) implies an intersection of the corresponding f-orbit with $Fix R \cap Σ_{1}$ . Both cases are possible for one orbit $O (u^{*})$ , recall Lemma 3.5 and Corollary 3.6.

Finally we define $\begin{matrix} S_{μ, R} : = h_{μ} (S_{R}^{2}) . \end{matrix}$

From (3.17) and the $σ$ -invariance of $S_{R}^{2}$ , the set $S_{μ, R}$ is $Π_{μ}$ -invariant, and the systems $(S_{R}^{2}, σ)$ and $(S_{μ, R}, Π_{μ})$ are topologically conjugated. Since topological transitivity and denseness of periodic point are topological properties we find by the aforementioned result by Banks et al. [2] that the system $(S_{μ, R}, Π_{μ})$ is chaotic in the sense of Devaney.

In the following section we consider how this chaotic dynamics dissolves within the region (II) displayed in Figure 4.

3.2. Dissolution of shift dynamics – local bifurcations

In this section we exclusively consider N-periodic orbits $X_{N} (μ)$ , $N \in {1, 2, 3, 4}$ . Of course for $μ$ within the region (I) the intersections of those orbits with $Σ_{2}$ belong to $S_{μ}$ . The aim of this section is to discuss how those orbits disappear within the region (II) by decreasing $μ_{2}$ .

3.2.1. Proof of Theorem 1.2

We proceed as in the Section 3.1 up to (3.5) which we repeat here $\begin{matrix} 0 = μ_{2} - u_{i}^{2} - \frac{2 c (u_{i}, μ)}{a (0, 0)} μ_{1}^{δ^{s}} + O (μ_{1}^{δ δ^{s}}), δ > 1, i \in Z . \end{matrix}$

Due to the sign condition (3.3), a(0, 0) and c(0, 0) are negative and since $c (u_{i}, μ)$ is bounded (for sufficiently small $u_{i}$ and $μ$ ), we have for sufficiently small $μ_{1}$ $\begin{matrix} - \frac{2 c (u_{i}, μ)}{a (0, 0)} μ_{1}^{δ^{s}} + O (μ_{1}^{δ δ^{s}}) < 0 . \end{matrix}$

This observation leads immediately to the statement of Theorem 1.2.

3.2.2. One-periodic orbits – proof of Theorem 1.3

Similar to [18], one-periodic orbits are characterised by sequences $ω = {(ω_{1, i}, ω_{2, i})}_{i \in Z}$ and $u = {(u_{i})}_{i \in Z}$ with $\begin{matrix} (ω_{1, i}, ω_{2, i}) = : (ω_{1}, ω_{2}), u_{i} = : u, i \in Z . \end{matrix}$

Therefore, according to (2.32) the bifurcation equations for 1-periodic orbits reads:(3.21) $\begin{matrix} 0 & = μ_{1} + a (u, μ) e^{- 2 λ^{u} ω_{1}} + R_{1}^{1} (u, ω, μ) \\ 0 & = - μ_{1} - a (u, μ) e^{- 2 λ^{u} ω_{2}} + R_{1}^{2} (u, ω, μ) \\ 0 & = μ_{2} - u^{2} + c (u, μ) e^{2 λ^{s} ω_{1}} + c (u, μ) e^{2 λ^{s} ω_{2}} + R_{2} (u, ω, μ) . \end{matrix}$ (3.21)

Define in the same manner as in (3.1) $\begin{matrix} r_{i} : = e^{- 2 λ^{u} ω_{i}}, i = 1, 2 . \end{matrix}$

Then there exists a $δ^{s} > 1$ such that the bifurcation equations in the new variables read(3.22) $\begin{matrix} 0 & = μ_{1} + a (u, μ) r_{1} + {\hat{R}}_{1}^{1} (u, r_{1}, r_{2}, μ) \\ 0 & = - μ_{1} - a (u, μ) r_{2} + {\hat{R}}_{1}^{2} (u, r_{1}, r_{2}, μ) \\ 0 & = μ_{2} - u^{2} + c (u, μ) r_{1}^{δ^{s}} + c (u, μ) r_{2}^{δ^{s}} + {\hat{R}}_{2} (u, r_{1}, r_{2}, μ) . \end{matrix}$ (3.22)

In what follows we assume as for the proof of Theorem 1.1 the sign condition (3.3).

The first two equations in (3.22) can be solved for $(r_{1}, r_{2}) (u, μ)$ ; because of (3.3) $μ_{1}$ has to be positive. After plugging in into the third equation in (3.22), this one can be solved for $μ_{2}$ :(3.23) $\begin{matrix} μ_{2} = u^{2} + {\hat{C}}_{1} (u, μ_{1}) . \end{matrix}$ (3.23)

The right hand side of (3.23) has a unique minimum $u^{*} (μ_{1})$ . Moreover, $u^{*} (μ_{1}) \to 0$ , as $μ_{1} \to 0$ . The above arguments prove Theorem 1.3 with $\begin{matrix} κ_{s c} (μ_{1}) : = (u^{*} (μ_{1}))^{2} + {\hat{C}}_{1} (u^{*} (μ_{1}), μ_{1}) = C μ_{1}^{δ^{s}} + o (μ_{1}^{δ^{s}}), \end{matrix}$

where, due to (3.3), C is a positive constant.

Lemma 3.14:

Let $μ_{1} > 0$ , let $μ_{2} \geq κ_{s c} (μ_{1})$ , and finally let u and $(ω_{1}, ω_{2})$ be values for which $X (u, (ω_{1}, ω_{2}), μ)$ is a 1-periodic orbit according to Theorem 1.3. Then $R X = X$ , i.e. X is symmetric.

ProofThe statement is an immediate consequence of Hypothesis (H8), Lemma 2.13 and the fact that, according to the analysis of (3.21) or (3.22), respectively, $(ω_{1}, ω_{2})$ is uniquely determined by u and $μ$ .

3.2.3. Two-periodic orbits – proof of Theorem 1.4

Two-periodic orbits are characterised by sequences $ω = {(ω_{1, i}, ω_{2, i})}_{i \in Z}$ and $u = {(u_{i})}_{i \in Z}$ with(3.24) $\begin{matrix} (ω_{1, i}, ω_{2, i}) = \{\begin{matrix} (ω_{1, 1}, ω_{2, 1}), & i odd \\ (ω_{1, 2}, ω_{2, 2}), & i even \end{matrix}, u_{i} = \{\begin{matrix} u_{1}, & i odd \\ u_{2}, & i even \end{matrix} . \end{matrix}$ (3.24)

We refer to Figure 7 for visualisation.

Figure 7. A two-periodic Lin orbit.

Display full size

With the setting of (3.1) there exists a $δ^{s} > 1$ such that the bifurcation equations for two-periodic orbits reads.(3.25) $\begin{matrix} 0 & = μ_{1} + a (u_{2}, μ) r_{1, 2} + {\hat{R}}_{1, 1}^{1} (u, r, μ) \\ 0 & = - μ_{1} - a (u_{1}, μ) r_{2, 1} + {\hat{R}}_{1, 1}^{2} (u, r, μ) \\ 0 & = μ_{2} - u_{1}^{2} + c (u_{1}, μ) r_{1, 1}^{δ^{s}} + c (u_{1}, μ) r_{2, 1}^{δ^{s}} + {\hat{R}}_{2, 1} (u, r, μ) \\ 0 & = μ_{1} + a (u_{1}, μ) r_{1, 1} + {\hat{R}}_{1, 2}^{1} (u, r, μ) \\ 0 & = - μ_{1} - a (u_{2}, μ) r_{2, 2} + {\hat{R}}_{1, 2}^{2} (u, r, μ) \\ 0 & = μ_{2} - u_{2}^{2} + c (u_{2}, μ) r_{1, 2}^{δ^{s}} + c (u_{2}, μ) r_{2, 2}^{δ^{s}} + {\hat{R}}_{2, 2} (u, r, μ) . \end{matrix}$ (3.25)

Now we proceed in principle as in the previous sections. We solve the subsystem consisting of the first two equations of each block in (3.25) for $\begin{matrix} r = (r_{1, 2}, r_{2, 1}, r_{1, 1}, r_{2, 2}) (u, μ) . \end{matrix}$

Plugging into the third equations of each block in (3.25) and solving these equations for $μ_{2}$ in each case yields two functions $μ_{2}^{k}$ , $k = 1, 2$ :(3.26) $\begin{matrix} μ_{2}^{1} = u_{1}^{2} + {\hat{C}}_{2, 1} (u, μ_{1}), μ_{2}^{2} = u_{2}^{2} + {\hat{C}}_{2, 2} (u, μ_{1}) . \end{matrix}$ (3.26)

Thus, the resulting determination equation for two-periodic orbits reads:(3.27) $\begin{matrix} μ_{2}^{1} = μ_{2}^{2} \Leftrightarrow 0 = u_{1}^{2} - u_{2}^{2} + {\hat{C}}_{2, 1} (u, μ_{1}) - {\hat{C}}_{2, 2} (u, μ_{1}) = : \hat{C} (u, μ_{1}) . \end{matrix}$ (3.27)

The right-hand side of the latter equation can be seen as a perturbation of $u_{1}^{2} - u_{2}^{2}$ . The zero set of $u_{1}^{2} - u_{2}^{2}$ consists of two intersecting straight lines. In what follows we show that, for fixed $μ_{1}$ , the solution set of the perturbation given in (3.27) still consists of two transversely intersecting curves. In the style of the shift operator $σ$ introduced in Section 3.1 we define the mapping $ζ$ on pairs (a, b): $\begin{matrix} ζ (a, b) = (b, a) . \end{matrix}$

Note that $ζ$ is related to the shift $σ$ on two-periodic sequences $(\bar{a b})$ .

Lemma 3.15:

$ζ ({\hat{C}}_{2, 1} (u, μ_{1}), {\hat{C}}_{2, 2} (u, μ_{1})) = ({\hat{C}}_{2, 1} (ζ u, μ_{1}), {\hat{C}}_{2, 2} (ζ u, μ_{1}))$ .

ProofThis statement follows from the uniqueness of Lin orbits (for given $u$ and $ω$ ). We also refer to the arguments used in the proof of Lemma 3.2.

An immediate consequence of Lemma 3.15 is ${\hat{C}}_{2, 1} (u, μ_{1}) = {\hat{C}}_{2, 2} (ζ u, μ_{1})$ , or in other words(3.28) $\begin{matrix} {\hat{C}}_{2, 1} (u_{1}, u_{2}, μ_{1}) = {\hat{C}}_{2, 2} (u_{2}, u_{1}, μ_{1}), \end{matrix}$ (3.28)

and hence(3.29) $\begin{matrix} {\hat{C}}_{2, 1} (u, u, μ_{1}) = {\hat{C}}_{2, 2} (u, u, μ_{1}) . \end{matrix}$ (3.29)

Therefore there is a function ${\hat{C}}_{r}$ such that we may write $\begin{matrix} \hat{C} (u, μ_{1}) = (u_{1} - u_{2}) (u_{1} + u_{2} + {\hat{C}}_{r} (u, μ_{1})) . \end{matrix}$

That means that one branch of two-periodic solutions exists for $u_{1} = u_{2}$ . Note that these solutions correspond to one-periodic orbits (which are passed through twice). So, the real two-periodic orbits (the ones with minimal period two) are related to solutions $u_{1} \neq u_{2}$ of(3.30) $\begin{matrix} 0 = u_{1} + u_{2} + {\hat{C}}_{r} (u, μ_{1}) . \end{matrix}$ (3.30)

It remains to show that the solutions of (3.30) form a curve (in $(u_{1}, u_{2})$ -space) intersecting the straight line ${u_{1} = u_{2}}$ transversely.

To this end we first note that $C_{i} (u, μ_{1})$ , $i = 1, 2$ , and their $u$ -derivatives are of order $O (μ_{1}^{δ^{s}})$ . Having that in mind we see that $\begin{matrix} 0 = 2 u + {\hat{C}}_{r} (u, u, μ_{1}) \end{matrix}$

has a solution $u = \hat{u} (μ_{1})$ , where $\hat{u} (μ_{1}) = O (μ_{1}^{δ^{s}})$ , and hence (3.30) has a solution $(u_{1}, u_{2}) = (\hat{u}, \hat{u})$ . Now we can apply the implicit function theorem to show that near $(\hat{u}, \hat{u})$ Equation (3.30) can be solved for $u_{2} = u_{2}^{*} (u_{1}, μ_{1})$ . It turns out that $D_{u_{1}} u_{2}^{*} (\hat{u}, μ_{1}) < 0$ . This finally shows that the solution curve of (3.30) indeed intersects the straight line ${u_{1} = u_{2}}$ transversely.

According to (3.26) and (3.27) we find that the function $κ_{p d} (μ_{1})$ (stated in Theorem 1.4) is defined by $\begin{matrix} κ_{p d} (μ_{1}) : = {\hat{u}}^{2} (μ_{1}) + {\hat{C}}_{2, 1} (\hat{u} (μ_{1}), \hat{u} (μ_{1}), μ_{1}) . \end{matrix}$

Similar to Lemma 3.14 we get

Lemma 3.16:

Let $μ_{1} > 0$ , let $μ_{2} \geq κ_{p d} (μ_{1})$ , and let $u$ and $ω$ , in accordance with (3.24) be values for which $X (u, ω, μ)$ is a 2-periodic orbit according to Theorem 1.3. Then $R X = X$ , i.e. X is symmetric.

ProofThe statement is, as the corresponding one of Lemma 3.14, an immediate consequence of the uniqueness: let $μ$ be fixed and let $(u_{1}, u_{2})$ , $u_{1} \neq u_{2}$ such that $X (u, ω, μ)$ is the corresponding two periodic orbit. Note that, in accordance with the above explanations $ω = ω (u, μ)$ , i.e. $ω$ is uniquely determined by $u$ and $μ$ . Furthermore, according to the uniqueness statements in Lemma 2.7(iv) and Theorem 2.9 we find that RX belongs to the same $u$ – and hence to the same $ω$ . Hence we have $R X = X$ .

Finally we comment on the size comparison of $κ_{p d} (μ_{1})$ and $κ_{s c} (μ_{1})$ . According to (3.28) we may write $\begin{matrix} {\hat{C}}_{2, 1} (u_{1}, u_{2}, μ_{1}) & = a_{0} + a_{1} u_{1} + a_{2} u_{2} + O ({(u_{1}, u_{2})}^{2}) \\ {\hat{C}}_{2, 2} (u_{1}, u_{2}, μ_{1}) & = a_{0} + a_{2} u_{1} + a_{1} u_{2} + O ({(u_{1}, u_{2})}^{2}) . \end{matrix}$

Note that $a_{i} = a_{i} (μ_{1})$ , $a_{i} (μ_{1}) = O (μ_{1}^{δ^{s}})$ . With that $\begin{matrix} {\hat{C}}_{r} (u, u, μ_{1}) = a_{1} - a_{2} + O (u^{2}), \end{matrix}$

and (3.30) can written as $\begin{matrix} u = \frac{a_{2} - a_{1}}{2} + O (u^{2}) . \end{matrix}$

Hence $\begin{matrix} \hat{u} (μ_{1}) = \frac{a_{2} - a_{1}}{2} + O (μ_{1}^{2 δ^{s}}), a_{i} (μ_{1}) = O (μ_{1}^{δ^{s}}) . \end{matrix}$

Next we compute the minimum $u^{*} (μ_{1})$ of the right-hand side of (3.23) in terms of $a_{i}$ . By construction we have $\begin{matrix} {\hat{C}}_{1} (u, μ_{1}) = {\hat{C}}_{2, 1} (u, u, μ_{1}) = (a_{1} + a_{2}) u + O (u^{2}) . \end{matrix}$

With that we find $\begin{matrix} u^{*} (μ_{1}) = - \frac{a_{2} + a_{1}}{2} + O (μ_{1}^{2 δ^{s}}), a_{i} (μ_{1}) = O (μ_{1}^{δ^{s}}) . \end{matrix}$

Therefore(3.31) $\begin{matrix} a_{2} \neq 0 \Rightarrow \hat{u} (μ_{1}) \neq u^{*} (μ_{1}) . \end{matrix}$ (3.31)

Now, in accordance with the definitions of $κ_{p d}$ and $κ_{s c}$ we find that (3.31) implies $\begin{matrix} κ_{p d} (μ_{1}) \neq κ_{s c} (μ_{1}) . \end{matrix}$

Figure 8. Let $μ_{1}$ be fixed. Left: solution branches related to two-periodic orbits – the solutions of (3.27). The straight line ${u_{1} = u_{2}}$ is related to one-periodic orbits which are passed through twice. The graph of $u_{2}^{*}$ is related to actual two-periodic orbits. Right: the graph of the function $u^{2} + {\hat{C}}_{1} (u, μ_{1})$ explains the saddle centre bifurcation of one-periodic orbits.

Display full size

Figure 8 explains that $a_{2} \neq 0$ more precisely implies $\begin{matrix} κ_{p d} (μ_{1}) > κ_{s c} (μ_{1}) . \end{matrix}$

3.2.4. 3-periodic orbits – the proof of Theorem 1.5

In this section we study symmetric 3-periodic orbits. Similar to the considerations in the previous subsections 3-periodic orbits are characterised by sequences $ω = {(ω_{1, i}, ω_{2, i})}_{i \in Z}$ and $u = {(u_{i})}_{i \in Z}$ with $\begin{matrix} (ω_{1, i + 3}, ω_{2, i + 3}) = (ω_{1, i}, ω_{2, i}), u_{i + 3} = u_{i}, i \in Z . \end{matrix}$

Due to Lemma 2.16, symmetric 3-periodic orbits are characterised, up to permutations, by the following restrictions (see also Figure 9):(3.32) $\begin{matrix} \begin{matrix} ω_{1, 1} = ω_{2, 1} \\ ω_{1, 2} = ω_{2, 3} \\ ω_{1, 3} = ω_{2, 2} \end{matrix} u_{2} = u_{3} . \end{matrix}$ (3.32)

Remark 3.17:

Note that the second lower indices 1, 2, 3 of $ω$ or the indices of u, respectively, correspond to $i = 0, 1, 2$ in the notation of Lemma 2.16. This implies that $i_{0}$ in the present setting is equal to zero.

Figure 9. A symmetric 3-periodic Lin orbit.

Display full size

The uniqueness statement in Theorem 2.9 implies (see again Figure 9 for a visualisation):

Lemma 3.18:

(3.33) $\begin{matrix} \begin{matrix} Ξ_{2, 2} & = 0 & \Leftrightarrow & Ξ_{2, 3} & = 0 \\ Ξ_{1, 1} & = 0 & \Leftrightarrow & Ξ_{1, 3} & = 0 \\ ⟨ Ξ_{1, 2}, ζ_{1}^{1} ⟩ & = 0 & \Leftrightarrow & ⟨ Ξ_{1, 2}, ζ_{1}^{2} ⟩ & = 0 . \end{matrix} \end{matrix}$ (3.33)

ProofThe statement follows mainly by means of Corollary 2.17. This immediately yields the first two equivalence statements in (3.33), recall Remark 3.17 in this respect. Also it gives(3.34) $\begin{matrix} R Ξ_{1, 2} = - Ξ_{1, 2} . \end{matrix}$ (3.34)

Furthermore, write $Ξ_{1, 2} = ⟨ Ξ_{1, 2}, ζ_{1}^{1} ⟩ ζ_{1}^{1} + ⟨ Ξ_{1, 2}, ζ_{1}^{2} ⟩ ζ_{1}^{2}$ . Exploiting $R ζ_{1}^{1} = ζ_{1}^{2}$ we see that (3.34) implies the third equivalence in (3.33).

According to (3.32) and (3.33) the set of equations determining symmetric 3-periodic orbit reduces to $\begin{matrix} Ξ_{1, 1} = 0, ⟨ Ξ_{1, 2}, ζ_{1}^{1} ⟩ = 0, Ξ_{2, 1} = 0, Ξ_{2, 2} = 0 . \end{matrix}$

Using the notation introduced in (3.1) this set of equations can be written:(3.35) $\begin{matrix} 0 & = μ_{1} + a (u_{2}, μ) r_{1, 2} + {\hat{R}}_{1, 1}^{1} (u, r, μ) \\ 0 & = - μ_{1} - a (u_{1}, μ) r_{1, 1} + {\hat{R}}_{1, 1}^{2} (u, r, μ) \\ 0 & = μ_{1} + a (u_{2}, μ) r_{1, 3} + {\hat{R}}_{1, 2}^{1} (u, r, μ) \\ 0 & = μ_{2} - u_{1}^{2} + 2 c (u_{1}, μ) r_{1, 1}^{δ^{s}} + {\hat{R}}_{2, 1} (u, r, μ) \\ 0 & = μ_{2} - u_{2}^{2} + c (u_{2}, μ) r_{1, 2}^{δ^{s}} + c (u_{2}, μ) r_{1, 3}^{δ^{s}} + {\hat{R}}_{2, 2} (u, r, μ) . \end{matrix}$ (3.35)

The first three equations in (3.35) can be solved for $\begin{matrix} r : = (r_{1, 1}, r_{1, 2}, r_{1, 3}) (u_{1}, u_{2}, μ) . \end{matrix}$

For the next few steps we proceed as in Section 3.2.3. The last two equations in (3.35) can be solved for $μ_{2}$ in each case, which yields two functions $μ_{2}^{k}$ , $k = 1, 2$ (3.36) $\begin{matrix} μ_{2}^{1} = u_{1}^{2} + {\tilde{C}}_{3, 1} (u_{1}, u_{2}, μ_{1}), μ_{2}^{2} = u_{2}^{2} + {\tilde{C}}_{3, 2} (u_{1}, u_{2}, μ_{1}) . \end{matrix}$ (3.36)

Thus, the resulting determination equation for symmetric 3-periodic orbits reads:(3.37) $\begin{matrix} μ_{2}^{1} = μ_{2}^{2} \Leftrightarrow 0 = u_{1}^{2} - u_{2}^{2} + {\tilde{C}}_{3, 1} (u_{1}, u_{2}, μ_{1}) - {\tilde{C}}_{3, 2} (u_{1}, u_{2}, μ_{1}) = : \tilde{C} (u_{1}, u_{2}, μ_{1}) . \end{matrix}$ (3.37)

Lemma 3.19:

${\tilde{C}}_{3, 1} (u, u, μ_{1}) = {\tilde{C}}_{3, 2} (u, u, μ_{1})$

ProofThis statement can be seen as an equivalent of (3.29). Indeed, it can be proved in a very similar way. To this end, for a start we disregard the symmetry condition (3.32) and consider the full bifurcation equation for 3-periodic orbits. If we proceed as in Section 3.2.3 we get corresponding functions ${\hat{C}}_{3, i} (u_{1}, u_{2}, u_{3}, μ_{1})$ , $i = 1, 2, 3$ . For these function an equivalent to Lemma 3.15 holds true. From that we find, by taking the symmetry condition (3.32) into account, the statement of the lemma. Note in this respect that ${\tilde{C}}_{3, i} (u_{1}, u_{2}, μ_{1}) = {\hat{C}}_{3, i} (u_{1}, u_{2}, u_{2}, μ_{1})$ , $i = 1, 2$ .

Indeed, Lemma 3.19 is a key point in our argumentation. Namely with that lemma we find $\begin{matrix} \tilde{C} (u, μ_{1}) = (u_{1} - u_{2}) (u_{1} + u_{2} + {\tilde{C}}_{r} (u, μ_{1})) . \end{matrix}$

With that we can proceed as in Section 3.2.3. Eventually we get a function $\tilde{u} (μ_{1})$ which solves $0 = 2 u + {\tilde{C}}_{r} (u, u, μ_{1})$ . Finally, near $(u_{1}, u_{2}) = (\tilde{u}, \tilde{u})$ , we can solve $0 = u_{1} + u_{2} + {\tilde{C}}_{r} (u, μ_{1})$ for $u_{2} = {\tilde{u}}_{2}^{*} (u_{1}, μ_{1})$ . The function $κ_{3 s h}$ stated in Theorem 1.5 is defined by $\begin{matrix} κ_{3 s h} (μ_{1}) : = {\tilde{u}}^{2} (μ_{1}) + {\tilde{C}}_{3, 1} (\tilde{u} (μ_{1}), \tilde{u} (μ_{1}), μ_{1}) . \end{matrix}$

3.2.5. 4-periodic orbits – the proof of Theorem 1.6

We may expect that symmetric periodic orbits which are related to chaotic dynamics, or in other words which are related to $S_{R}^{2}$ , can be continued into region (II) (cf. Figure 4). Consider the periodic orbits in $S_{R}^{2}$ with minimal period four. These correspond to the 4-periodic sequences $(\bar{+ - - -})$ , $(\bar{+ + - -})$ and $(\bar{+ + + -})$ . We denote the corresponding f-orbits by $O_{1 +}$ , $O_{2 +}$ and $O_{3 +}$ respectively. In accordance with Remark 3.13 we find $O_{1 +}$ and $O_{3 +}$ have two intersections with $Fix R \cap Σ_{2}$ in each case while $O_{2 +}$ has no intersection with $Fix R \cap Σ_{2}$ . Similarly we find that the 1-periodic f-orbits corresponding to $(\bar{+})$ or $(\bar{-})$ have exactly one intersection with $Fix R \cap Σ_{2}$ in each case and the 2-periodic f-orbit corresponding to $(\bar{+ -})$ has two intersections with $Fix R \cap Σ_{2}$ . We refer to Figure 10 for a visualisation.

Figure 10. Possible subharmonic bifurcations from 1- or 2-periodic orbits to 4-periodic orbits. Displayed are the intersections of symmetric periodic orbits (black dots) with $Fix R$ (vertical lines).

Display full size

Now, consider 4-periodic f-orbits as depicted in Figure 10 which exist in region (I), cf. Figure 4. If we continue those orbits into region (II) then we find, similar to Section 3.2.4, the following: the continuation of $O_{2 +}$ (depicted in panel (a) of Figure 10) is related to a 4-periodic $u$ -sequence with $u_{1} = u_{2}$ and $u_{3} = u_{4}$ . Similarly, the continuations of both $O_{1 +}$ and $O_{3 +}$ (depicted in the panels (b) and (c) of Figure 10), are related to a 4-periodic $u$ -sequence with $u_{2} = u_{4}$ . We emphasise that, without further examinations or assumptions, it is impossible to decide which of the orbits $O_{1 +}$ and $O_{3 +}$ bifurcates from a 1-periodic orbit and which one bifurcates from the branch of 2-periodic orbits.

First we consider the continuation of $O_{2 +}$ and show that this, as suggested by Figure 10 panel (a), bifurcates from a branch of 1-periodic orbits. This proves part of Theorem 1.6(i), namely that one branch of 4-periodic orbits bifurcates from the branch of 1-periodic orbits. However from general theory it is known that in the course of a subharmonic bifurcation two branches bifurcate, cf. the discussion below. In the second part of this section we will also discuss how it can be shown that the continuation of one of the orbits $O_{1 +}$ and $O_{3 +}$ will bifurcate from the branch of 1-periodic orbits.

By a sketch similar to the one in Figure 9 it becomes clear that the continuation of a symmetric periodic orbit $O_{2 +}$ is related to sequences $ω$ and $u$ with $\begin{matrix} \begin{matrix} ω_{1, 1} = ω_{2, 2}, ω_{1, 2} = ω_{2, 1}, ω_{1, 3} = ω_{2, 4}, ω_{1, 4} = ω_{2, 3}, \\ u_{1} = u_{2}, u_{3} = u_{4} . \end{matrix} \end{matrix}$

Furthermore, as the counterpart of Lemma 3.18 we find $\begin{matrix} \begin{matrix} Ξ_{1, 2} & = 0 & \Leftrightarrow & Ξ_{1, 4} & = 0, \\ ⟨ Ξ_{1, 1}, ζ_{1}^{1} ⟩ & = 0 & \Leftrightarrow & ⟨ Ξ_{1, 1}, ζ_{1}^{2} ⟩ & = 0, \\ ⟨ Ξ_{1, 3}, ζ_{1}^{1} ⟩ & = 0 & \Leftrightarrow & ⟨ Ξ_{1, 3}, ζ_{1}^{2} ⟩ & = 0, \end{matrix} \begin{matrix} Ξ_{2, 1} & = 0 & \Leftrightarrow & Ξ_{2, 2} & = 0, \\ Ξ_{2, 3} & = 0 & \Leftrightarrow & Ξ_{2, 4} & = 0 . \end{matrix} \end{matrix}$

With that we proceed as in Section 3.2.4. In doing so we arrive at the counterpart of (3.36) $\begin{matrix} μ_{2}^{1} = u_{1}^{2} + {\tilde{C}}_{4, 1} (u_{1}, u_{3}, μ_{1}), μ_{2}^{3} = u_{3}^{2} + {\tilde{C}}_{4, 3} (u_{1}, u_{3}, μ_{1}) . \end{matrix}$

Thus, the resulting determination equation for symmetric 4-periodic orbits reads: $\begin{matrix} μ_{2}^{1} = μ_{2}^{3} \Leftrightarrow 0 = u_{1}^{2} - u_{3}^{2} + {\tilde{C}}_{4, 1} (u_{1}, u_{3}, μ_{1}) - {\tilde{C}}_{4, 3} (u_{1}, u_{3}, μ_{1}) = : \tilde{C} (u_{1}, u_{3}, μ_{1}) . \end{matrix}$

Parallel to Lemma 3.19 we find $\begin{matrix} {\tilde{C}}_{4, 1} (u, u, μ_{1}) = {\tilde{C}}_{4, 3} (u, u, μ_{1}) . \end{matrix}$

This enables us to proceed along the lines of the remaining part of Section 3.2.4 and we finally obtain $\begin{matrix} κ_{4 s h} (μ_{1}) : = {\tilde{u}}^{2} (μ_{1}) + {\tilde{C}}_{4, 1} (\tilde{u} (μ_{1}), \tilde{u} (μ_{1}), μ_{1}), \end{matrix}$

where $\tilde{u} (\cdot)$ is defined in the same way as in Section 3.2.4. However we note that the function $\tilde{u} (\cdot)$ used here does not coincide with one used in Section 3.2.4.

Next we consider continuations of the orbits $O_{1 +}$ and $O_{3 +}$ . For those orbits we find that they are related to sequences $ω$ and $u$ with $\begin{matrix} ω_{1, 1} = ω_{2, 1}, ω_{1, 2} = ω_{2, 4}, ω_{1, 3} = ω_{2, 3}, ω_{1, 4} = ω_{2, 2}, and u_{2} = u_{4} . \end{matrix}$

For the jumps $Ξ_{i, j}$ we find $\begin{matrix} \begin{matrix} Ξ_{1, 1} & = 0 & \Leftrightarrow & Ξ_{1, 4} & = 0, \\ Ξ_{1, 2} & = 0 & \Leftrightarrow & Ξ_{1, 3} & = 0, \end{matrix} \begin{matrix} Ξ_{2, 2} & = 0 & \Leftrightarrow & Ξ_{2, 4} & = 0 . \end{matrix} \end{matrix}$

We again proceed as in Section 3.2.4. The counterpart of (3.36) reads $\begin{matrix} μ_{2}^{1} & = u_{1}^{2} + {\tilde{C}}_{4, 1} (u_{1}, u_{2}, u_{3}, μ_{1}), μ_{2}^{2} = u_{2}^{2} + {\tilde{C}}_{4, 2} (u_{1}, u_{2}, u_{3}, μ_{1}), \\ μ_{2}^{3} & = u_{3}^{2} + {\tilde{C}}_{4, 3} (u_{1}, u_{2}, u_{3}, μ_{1}), \end{matrix}$

whereas the counterpart to (3.37) is given by(3.38) $\begin{matrix} \begin{matrix} μ_{2}^{1} = μ_{2}^{2} & \Leftrightarrow & 0 = u_{1}^{2} - u_{2}^{2} + {\tilde{C}}_{4, 1} (u_{1}, u_{2}, u_{3}, μ_{1}) - {\tilde{C}}_{4, 2} (u_{1}, u_{2}, u_{3}, μ_{1}) \\ μ_{2}^{2} = μ_{2}^{3} & \Leftrightarrow & 0 = u_{3}^{2} - u_{2}^{2} + {\tilde{C}}_{4, 3} (u_{1}, u_{2}, u_{3}, μ_{1}) - {\tilde{C}}_{4, 2} (u_{1}, u_{2}, u_{3}, μ_{1}) . \end{matrix} \end{matrix}$ (3.38)

Define(3.39) $\begin{matrix} {\tilde{C}}_{4, 1} - {\tilde{C}}_{4, 2} = : {\tilde{C}}_{412} and {\tilde{C}}_{4, 3} - {\tilde{C}}_{4, 2} = : {\tilde{C}}_{432} \end{matrix}$ (3.39)

Lemma 3.20:

${\tilde{C}}_{412} (u_{1}, u_{2}, u_{3}) = {\tilde{C}}_{432} (u_{3}, u_{2}, u_{1})$ .

ProofUsing arguments similar to that given in the proof of Lemma 3.19 we find: $\begin{matrix} {\tilde{C}}_{412} (u_{1}, u_{2}, u_{3}) & = {\tilde{C}}_{4, 1} (u_{1}, u_{2}, u_{3}) \\ - {\tilde{C}}_{4, 2} (u_{1}, u_{2}, u_{3}) = {\hat{C}}_{4, 1} (u_{1}, u_{2}, u_{3}, u_{2}) - {\hat{C}}_{4, 2} (u_{1}, u_{2}, u_{3}, u_{2}) \\ = {\hat{C}}_{4, 3} (u_{3}, u_{2}, u_{1}, u_{2}) - {\hat{C}}_{4, 2} (u_{3}, u_{2}, u_{1}, u_{2}) \\ = {\tilde{C}}_{432} (u_{3}, u_{2}, u_{1}) . \end{matrix}$

Using the notation introduced in (3.39), system (3.38) can be written as(3.40) $\begin{matrix} F (u_{1}, u_{2}, u_{3}, μ_{1}) = 0, F (u_{1}, u_{2}, u_{3}, μ_{1}) : = (\begin{matrix} u_{1}^{2} - u_{2}^{2} + {\tilde{C}}_{412} (u_{1}, u_{2}, u_{3}, μ_{1}) \\ u_{3}^{2} - u_{2}^{2} + {\tilde{C}}_{432} (u_{1}, u_{2}, u_{3}, μ_{1}) \end{matrix}) . \end{matrix}$ (3.40)

Since 1-periodic solution also are covered by the equation $F (u_{1}, u_{2}, u_{3}, μ_{1}) = 0$ we find(3.41) $\begin{matrix} F (u, u, u, μ_{1}) = 0 . \end{matrix}$ (3.41)

Hence, by means of the Mean Value Theorem we find that F can be written as(3.42) $\begin{matrix} F (u_{1}, u_{2}, u_{3}, μ_{1}) = (\begin{matrix} F_{1}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) \\ F_{1}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) \end{matrix}) (\begin{matrix} u_{1} - u_{2} \\ u_{3} - u_{2} \end{matrix}) . \end{matrix}$ (3.42)

Furthermore, the functions $F_{i}^{j}$ are related to the partial derivative of F with respect to $(u_{1}, u_{3})$ . More precisely $\begin{matrix} (\begin{matrix} F_{1}^{1} (u, u, u, μ_{1}) & F_{3}^{1} (u, u, u, μ_{1}) \\ F_{1}^{2} (u, u, u, μ_{1}) & F_{3}^{2} (u, u, u, μ_{1}) \end{matrix}) & = (\begin{matrix} 2 u + D_{1} {\tilde{C}}_{412} (u, u, u, μ_{1}) & D_{3} {\tilde{C}}_{412} (u, u, u, μ_{1}) \\ D_{1} {\tilde{C}}_{432} (u, u, u, μ_{1}) & 2 u + D_{3} {\tilde{C}}_{432} (u, u, u, μ_{1}) \end{matrix}) \\ = (\begin{matrix} 2 u + D_{1} {\tilde{C}}_{412} (u, u, u, μ_{1}) & D_{3} {\tilde{C}}_{412} (u, u, u, μ_{1}) \\ D_{3} {\tilde{C}}_{412} (u, u, u, μ_{1}) & 2 u + D_{1} {\tilde{C}}_{412} (u, u, u, μ_{1}) \end{matrix}), \end{matrix}$

where the last equality follows from Lemma 3.20. Applying the implicit function theorem we obtain the following:

Lemma 3.21:

The equation $2 u + D_{1} {\tilde{C}}_{412} (u, u, u, μ_{1}) - D_{3} {\tilde{C}}_{412} (u, u, u, μ_{1}) = 0$ has a unique solution $\hat{u}$ .

Now we reconsider the equation $F (u_{1}, u_{2}, u_{3}, μ_{1}) = 0$ . To find solutions which are related to proper 4-periodic solutions we consider the system(3.43) $\begin{matrix} \begin{matrix} F^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) & : = det (\begin{matrix} F_{1}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) \\ F_{1}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) \end{matrix}) = 0 \\ F^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) & : = ⟨ (F_{1}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}), F_{3}^{1} (u_{1}, u_{2}, u_{3}, μ_{1})), (u_{1} - u_{2}, u_{3} - u_{2}) ⟩ = 0 . \end{matrix} \end{matrix}$ (3.43)

We show that near $(u_{1}, u_{2}, u_{3}) = (\hat{u}, \hat{u}, \hat{u})$ the system () can be solved for $(u_{1}, u_{2}) (u_{3})$ , and since $\hat{u}$ is an isolated zero of $F^{1} (\hat{u}, \hat{u}, \hat{u})$ we find that $(u_{1}, u_{2}) (u_{3}) \neq (u_{3}, u_{3})$ .

Due to Lemma 3.21 it is clear that $F^{1} (\hat{u}, \hat{u}, \hat{u}) = 0$ , and by its definition it is immediately clear that $F^{2} (\hat{u}, \hat{u}, \hat{u}) = 0$ . It can easily be verified that ${\frac{\partial (F^{1}, F^{2})}{\partial (u_{1}, u_{2})}|}_{(\hat{u}, \hat{u}, \hat{u})}$ is non-singular. With the implicit function theorem this gives the solution $(u_{1}, u_{2}) (u_{3})$ .

In a similar way we can handle the period doubling from 2- to 4-periodic orbits. According to Section 3.2.3 all 2-periodic orbits are symmetric. These orbits intersect $Fix R \cap Σ_{2}$ twice (see also Figure 8 panel (c)). Let $u_{2 p, 1}$ and $u_{2 p, 2}$ be the u-components of the intersection points. We have $u_{2 p, 1} \neq u_{2 p, 2}$ . Finally we know that, for fixed $μ_{1}$ , there exists a function $u_{2}^{*}$ such that the graph $(u_{2 p, 1}, u_{2}^{*} (u_{2 p, 1}))$ describes the branch of 2-periodic orbits $(u_{2 p, 1} (μ_{2}), u_{2 p, 2} (μ_{2}))$ in the $(u_{1}, u_{2})$ -plane, see also Figure 8.

Equation (3.40) also covers the branch of 2-periodic solutions. So we find as the counterpart of (3.41) $\begin{matrix} F (u_{2}^{*} (u_{2}), u_{2}, u_{2}^{*} (u_{2})) \equiv 0 . \end{matrix}$

In the same way as explained above this leads to the representation of F $\begin{matrix} F (u_{1}, u_{2}, u_{3}, μ_{1}) = (\begin{matrix} F_{1}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{1} (u_{1}, u_{2}, u_{3}, μ_{1}) \\ F_{1}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) & F_{3}^{2} (u_{1}, u_{2}, u_{3}, μ_{1}) \end{matrix}) (\begin{matrix} u_{1} - u_{2}^{*} (u_{2}) \\ u_{3} - u_{2}^{*} (u_{2}) \end{matrix}) . \end{matrix}$

Note that $F_{i}^{j}$ are not the same as the ones in (3.42).

Further, similar to Lemma 3.21 we can show that the equation $\begin{matrix} F_{1}^{1} (u_{2}^{*} (u_{2}), u_{2}, u_{2}^{*} (u_{2})) + F_{1}^{3} (u_{2}^{*} (u_{2}), u_{2}, u_{2}^{*} (u_{2})) = 0 \end{matrix}$

has a unique solution $\hat{u}$ (which is also different from the one stated in Lemma 3.21). By symmetry arguments, see Lemma 3.20, we find that $F (u_{2}^{*} (u_{2}), u_{2}, u_{2}^{*} (u_{2}))$ has rank one. This allows to proceed in exactly the same way as above from Equation () on.

4. Discussion

Based on our considerations in the previous section or Theorems 1.3–1.6, respectively, it seems a likely supposition that the transition from no recurrent dynamics to shift dynamics is mainly governed by saddle-centre bifurcations, period doubling bifurcations and subharmonic bifurcations. In Figure 11 we present a part of a possible bifurcation diagram to explain the bifurcations of symmetric periodic orbits in the region (II) (of Figure 4) while taking our results into account. We emphasise that our focus here is just to describe a plausible mechanism by which the shift dynamics dissolves in the region (II), and we make no claim that our description of the dynamics in this region is complete. We expect that a general description of the dynamics in this region is considerably more complicated.

Figure 11. Possible bifurcation diagram for symmetric periodic orbits.

Display full size

Note that in Theorems 1.3–1.6 we only made statements concerning bifurcations of symmetric periodic orbits with period less than five. Since our proofs are based on Lin’s method we are not able to make any statements concerning the stability of the involved orbits. In particular, we cannot say anything about the Floquet multipliers (of those orbits) at the bifurcation points. However, the scenario depicted in Figure 11 is supported by studies of local bifurcations in generic (codimension one) families of vector fields. First note that the 2n-dimensional Poincaré map of a symmetric periodic orbit will inherit the reversing symmetry R, and therefore the Floquet multipliers will come in pairs that multiply to one. Generic codimension one bifurcations can therefore be reduced to a 2-dimensional centre manifold, where the dynamics on the centre manifold again inherits the reversing symmetry.

Consider first the cases where a pair of Floquet multipliers are equal to either 1 or $- 1$ . Denote the Poincaré map of the symmetric periodic orbit on the centre manifold by $L (\cdot, μ)$ , and suppose that $L (0, 0) = 0$ , and $D L (0, 0) = A_{s} + A_{n}$ is the canonical splitting of the Frechet derivative of L at (0, 0) into its semisimple and nilpotent part. Then it can be shown (see e.g. [23, Theorem 1.1]) that there exists an R-equivariant coordinate transformation $T_{μ}$ such that $\begin{matrix} T_{μ}^{- 1} \circ L \circ T_{μ} and A_{s} \circ χ_{τ} (μ) \end{matrix}$

have the same Taylor expansion up to an arbitrarily high order at (0, 0), where $χ_{τ} (μ)$ is the time-one map of the flow of a vector field $χ (μ)$ that commutes with $A_{s}$ and has R as a reversing symmetry.

When the Floquet multipliers are equal to 1, then $A_{s} = I$ and generically the vector field is determined by its 2-jet [24]: $\begin{matrix} \dot{x} & = y, \\ \dot{y} & = μ + a x^{2}, a \neq 0 . \end{matrix}$

The above normal form results in a saddle-centre bifurcation (see e.g. [24, Figure 1]), and the map L is equal (up to the R-equivariant coordinate transformation $T_{μ}$ ) to the time-one map $χ_{τ} (μ)$ up to arbitrarily high order. Therefore, the periodic orbit generically undergoes a saddle-centre bifurcation. We remark that addition of the higher order terms to the map L will generically break the invariant tori surrounding the elliptic periodic orbit resulting in chaotic dynamics.

When the Floquet multipliers are equal to $- 1$ , then $A_{s} = - I$ , and in this case the vector field has the generic unfolding (see again [24]): $\begin{matrix} \dot{x} & = μ y + a y^{3}, a \neq 0, \\ \dot{y} & = x . \end{matrix}$

The above normal form gives rise to a pitchfork bifurcation (see e.g. [24, Figure 3(b) and (c)]). However this time the map L is equivalent (up to arbitrarily high order) to $A_{s} \circ χ_{τ} (μ) = - χ_{τ} (μ)$ , and so the periodic orbit undergoes a period-doubling bifurcation. We remark that the bifurcating (period-doubled) orbit will be R-symmetric.

Furthermore, studies by Vanderbauwhede and Ciocci [4,5,29,30] describe the generic situation as Floquet multipliers travel around the unit circle, as depicted in the elliptic regions in Figure 11. These results are given in the following theorem.

Theorem 4.1:

[4, Theorem 3] Let $Φ_{λ} (\cdot)$ be a one-parameter family of reversible diffeomorphisms on $R^{2 n}$ , and let 0 be a symmetric fixed point of $Φ_{0}$ . Assume

(i)	$D Φ_{0} (0)$ has a pair of simple eigenvalues $e^{\pm \frac{2 π i p}{q}}$ , where $p, q \in N$ , $\gcd (p, q) = 1$ ,
(ii)	$D Φ_{0} (0)$ has no other eigenvalues which are root of unity,
(iii)	the continuation of the eigenvalue $e^{\frac{2 π i p}{q}}$ crosses the root of unity transversely (on the unit circle $S^{1}$ ) by moving $λ$ off zero.

Assume additionally that a further non-degeneracy condition (ND) is satisfied. Then, if

q \geq 3

, exactly two branches of symmetric q-periodic orbits bifurcate from the fixed point

x = 0

λ = 0

. Also, for

q \geq 5

the orbits in one branch are stable, while those in the other branch are unstable.

The ND mentioned in the theorem refers to a certain coefficient in the bifurcation equation. For more details, as well as for the proof we refer to [4]. But we note that due to the symmetry of the fixed point, there is a ‘symmetry’ for the set of eigenvalues of $D Φ_{0} (0)$ . This finally implies that under the assumption of the theorem $dim Fix R = dim Fix (- R)$ (where R is the involved involution). Finally we note that, although the theorem only refers to subharmonic bifurcations ( $q \geq 3$ ), by the same methods (as used in the proof of the theorem) the continuation/bifurcation of one- and two-period orbits can also be studied. This has been done for instance in [5] in a somewhat different context.

To explain the diagram in Figure 11 we perform the following thought experiment. Consider the two branches of one-periodic orbits which merges at $μ_{2} = κ_{s c} (μ_{1})$ . We assume that this happens in the course of a saddle-center bifurcation (note that we only proved the coalescence of the periodic orbits). Let us also assume that along one branch the periodic orbits have a pair of simple Floquet multipliers on the unit circle $S^{1}$ . Increasing $μ_{2}$ causes a motion of these multipliers along $S^{1}$ until they meet again at $- 1$ and the period doubling occurs. Along its way on $S^{1}$ these multipliers have to pass roots of unity. If this passage happens in accordance with assumptions of Theorem 4.1 then subharmonic bifurcations as suggested in Figure 11 will occur. If $q \geq 3$ then (again in accordance with Theorem 4.1) the periodic orbits in one branch are stable. In the reversible context that means that they have Floquet multipliers on $S^{1}$ . They will also move on $S^{1}$ as $μ_{1}$ changes and can cross roots of unity. If at those points the assumptions of Theorem 4.1 are met again corresponding subharmonic bifurcations from this q-periodic orbit will occur. This leads to a cascade of subharmonic bifurcations. We note that a simultaneous cascade of period doubling bifurcations takes place as has been described in [27, Section 5.2] for two-dimensional reversible maps. We also refer to [22, Section 3.2].

However, we cannot explain the disappearance (for decreasing $μ_{2}$ ) of all the periodic orbits which are involved in the shift dynamics only by means of such cascades of subharmonic and period doubling bifurcations. Namely, simple enumerative combinatorics reveals that there are six 5-periodic orbits. All of these are symmetric. But only four of them can be created (or disappear) in the course of subharmonic bifurcations from corresponding branch of one-periodic orbits. So we conjecture that the remaining two branches undergo a saddle-center bifurcation – cf. point $P_{s c}$ in Figure 11.

Acknowledgements

Finally, we would like to thank André Vanderbauwhede for providing the reference for Theorem 4.1 in personal communication.

Additional information

Funding

K.N. Webster was supported by a Marie Skl odowska-Curie Individual Fellowship [grant number 660616]. The authors furthermore gratefully acknowledge support through EPSRC Mathematics Platform Grant (JK); Russian Science Foundation [grant number 14-41-00044] at the Lobachevsky University of Nizhny Novgorod (JL); and EU Horizon 2020 ITN CRITICS [grant number 643073] (JL).

Notes

No potential conflict of interest was reported by the authors.

References

V.S. Afraimovich, V.V. Bykov, and L.P. Shilnikov, The origin and structure of the Lorenz attractor (Russian), Dokl. Akad. Nauk SSSR 234 (1977), pp. 336–339. [Google Scholar]

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Am. Math. Monthly 99 (1992), pp. 332–334. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]

V.V. Bykov, The bifurcations of separatrix contours and chaos, Physica D 62(1–4) (1993), pp. 290–299. [Crossref], [Google Scholar]

M.-C. Ciocci, Generalized Lyapunov–Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms, J. Differ. Equ. Appl. 10(7) (2004), pp. 621–649. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]

M.C. Ciocci and A. Vanderbauwhede, Bifurcation of periodic points in reversible diffeomorphisms, in New progress in difference equations, B. Aulbach, et al., ed., CRC Press, 2004, pp. 75–93. Proceedings of the 6th international conference on difference equations, Augsburg, Germany, July 30–August 3, 2001. [Google Scholar]

R. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc. 218 (1976), pp. 89–113. [Crossref], [Web of Science ®], [Google Scholar]

R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading, 1989. [Google Scholar]

J. Dugundji, Topology, Allyn and Bacon, Boston, 1987. [Google Scholar]

F. Dumortier, S. Ibáñez, and H. Kokubu, Cocoon bifurcation in three-dimensional reversible vector fields, Nonlinearity 19(2) (2006), pp. 305–328. [Crossref], [Web of Science ®], [Google Scholar]

F. Fernández-Sánchez, E. Freire, L. Pizarro, and A.J. Rodríguez-Luis, A model for the analysis of the dynamical consequences of a nontransversal intersection of the two-dimensional manifolds involved in a T-point, Phys. Lett. A. 320(2–3) (2003), pp. 169–179. [Crossref], [Web of Science ®], [Google Scholar]

F. Fernández-Sánchez, E. Freire, and A.J. Rodríguez-Luis, Bi-spiraling homoclinic curves around a T-point in Chua’s equation, Int. J. Bifur. Chaos Appl. Sci. Eng. 14 5(2004), pp. 1789–1793. [Crossref], [Web of Science ®], [Google Scholar]

P. Glendinning and C. Sparrow, T-points: A codimension two heteroclinic bifurcation, J. Statist. Phys. 43 (1986), pp. 479–488. [Crossref], [Web of Science ®], [Google Scholar]

A.J. Homburg, A.C. Jukes, J. Knobloch, and J.S.W. Lamb, Bifurcation from codimension one relative homoclinic cycles, Trans. Am. Math. Soc. 363(11) (2011), pp. 5663–5701. [Crossref], [Web of Science ®], [Google Scholar]

A.J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems Vol. 3, B. Hasselblatt and A. Katok, eds., North-Holland, Amsterdam, 2010, pp. 379–524. [Crossref], [Google Scholar]

J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Differ. Equ. 9(3) (1997), pp. 427–444. [Crossref], [Google Scholar]

J. Knobloch, Lin’s method for discrete and continuous dynamical systems and applications, Habilitation thesis, TU Ilmenau, 2004. Available at www.tu-ilmenau.de/fileadmin/media/analysis/knobloch/paper/lin04.pdf.gz. [Google Scholar]

J. Knobloch, Chaotic behaviour near non-transversal homoclinic points with quadratic tangency, J. Differ. Equ. Appl. 12(10) (2006), pp. 1037–1056. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]

J. Knobloch, J. Lamb, and K.N. Webster, Using Lin’s method to solve Bykov’s Problems, J. Differ. Equ. 257 (2014), pp. 2984–3047. [Crossref], [Web of Science ®], [Google Scholar]

I.S. Labouriau and A.A.P. Rodrigues, Global generic dynamics close to symmetry, J. Differ. Equ. 253 (2012), pp. 2527–2557. [Crossref], [Web of Science ®], [Google Scholar]

I.S. Labouriau and A.A.P. Rodrigues, Dense heteroclinic tangencies near a Bykov cycle, J. Differ. Equ. 259 (2015), pp. 5875–5902. [Crossref], [Web of Science ®], [Google Scholar]

I.S. Labouriau and A.A.P. Rodrigues, Global bifurcations close to symmetry, preprint (2015). Available at arXiv:1504.01659v1. [Google Scholar]

J.S.W. Lamb, Reversing symmetries in dynamical systems, Ph.D. thesis, University of Amsterdam1994. [Google Scholar]

J.S.W. Lamb, Local bifurcations in k-symmetric dynamical systems, Nonlinearity 9 (1996), pp. 537–558. [Crossref], [Web of Science ®], [Google Scholar]

J.S.W. Lamb and H.W. Capel, Local bifurcations on the plane with reversing point group symmetry, Chaos Solitons Fractals 5(2) (1995), pp. 271–293. [Crossref], [Web of Science ®], [Google Scholar]

J.S.W. Lamb and J.A.G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physica D 112(1–2) (1998), pp. 1–39. [Crossref], [Google Scholar]

J.S.W. Lamb, M.-A. Teixeira, and K.N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in ℝ³, J. Differ. Equ. 219(1) (2005), pp. 78–115. [Crossref], [Web of Science ®], [Google Scholar]

J.A.G. Roberts and G.R.W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216(2 &3) (1992), pp. 63–177. [Crossref], [Google Scholar]

B. Sandstede, Verzweigungtheorie homokliner Verdopplungen, PhD thesis, University of Stuttgart1993. Available at http://www.maths.surrey.ac.uk/personal/st/B.Sandstede/publications/dissertation.pdf [Google Scholar]

A. Vanderbauwhede, Bifurcation of subharmonic solutions in time-reversible systems, Z. Angew. Math. Phys. 37 (1986), pp. 455–478. [Crossref], [Web of Science ®], [Google Scholar]

A. Vanderbauwhede, Subharmonic branching in reversible systems, SIAM J. Math. Anal. 21(4) (1990), pp. 954–979. [Crossref], [Web of Science ®], [Google Scholar]

A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. 43(2) (1992), pp. 292–318. [Crossref], [Web of Science ®], [Google Scholar]

Journal of Difference Equations and Applications

Shift dynamics near non-elementary T-points with real eigenvalues

Special Section Dedicated to Andre Vanderbauwhede

Shift dynamics near non-elementary T-points with real eigenvalues

Abstract
Formulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

1. Introduction

Published online:

Published online:

Published online:

Published online:

2. Lin’s method

2.1. Splitting of the stable and unstable Manifolds

2.2. Construction of Lin orbits

Published online:

2.3. The determination equation for orbits near $Γ$

Published online:

3. Dynamics near $Γ$

3.1. Shift dynamics – Proof of Theorem 1.1

3.1.1. Symbolic dynamics

3.1.2. Reversible symbolic dynamics

3.2. Dissolution of shift dynamics – local bifurcations

3.2.1. Proof of Theorem 1.2

3.2.2. One-periodic orbits – proof of Theorem 1.3

3.2.3. Two-periodic orbits – proof of Theorem 1.4

Published online:

Published online:

3.2.4. 3-periodic orbits – the proof of Theorem 1.5

Published online:

3.2.5. 4-periodic orbits – the proof of Theorem 1.6

Published online:

4. Discussion

Published online:

Funding

Related research

Information for

Open access

Opportunities

Help and information

Journal of Difference Equations and Applications

Shift dynamics near non-elementary T-points with real eigenvalues

Special Section Dedicated to Andre Vanderbauwhede

Shift dynamics near non-elementary T-points with real eigenvalues

AbstractFormulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

1. Introduction

Published online:

Published online:

Published online:

Published online:

2. Lin’s method

2.1. Splitting of the stable and unstable Manifolds

2.2. Construction of Lin orbits

Published online:

2.3. The determination equation for orbits near <img src="/na101/home/literatum/publisher/tandf/journals/content/gdea20/2018/gdea20.v024.i04/10236198.2017.1331890/20180305/images/gdea_a_1331890_ilm0500.gif" alt="" />Γ

Published online:

3. Dynamics near <img src="/na101/home/literatum/publisher/tandf/journals/content/gdea20/2018/gdea20.v024.i04/10236198.2017.1331890/20180305/images/gdea_a_1331890_ilm0662.gif" alt="" />Γ

3.1. Shift dynamics – Proof of Theorem 1.1

3.1.1. Symbolic dynamics

3.1.2. Reversible symbolic dynamics

3.2. Dissolution of shift dynamics – local bifurcations

3.2.1. Proof of Theorem 1.2

3.2.2. One-periodic orbits – proof of Theorem 1.3

3.2.3. Two-periodic orbits – proof of Theorem 1.4

Published online:

Published online:

3.2.4. 3-periodic orbits – the proof of Theorem 1.5

Published online:

3.2.5. 4-periodic orbits – the proof of Theorem 1.6

Published online:

4. Discussion

Published online:

Acknowledgements

Additional information

Funding

Notes

References

Related research

Information for

Open access

Opportunities

Help and information

Keep up to date

Abstract
Formulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

2.3. The determination equation for orbits near $Γ$

3. Dynamics near $Γ$