Dynamic system with no equilibrium and its chaos anti-synchronization

ABSTRACT Recently, systems with chaos and the absence of equilibria have received a great deal of attention. In our work, a simple five-term system and its anti-synchronization are presented. It is special that the system has a hyperbolic sine nonlinearity and no equilibrium. Such a system generates chaotic behaviours, which are verified by phase portraits, positive Lyapunov exponent as well as an electronic circuit. Moreover, the system displays multistable characteristic when changing its initial conditions. By constructing an adaptive control, chaos anti-synchronization of the system with no equilibrium is obtained and illustrated via a numerical example.

Numerous chaotic systems with different terms were studied. Butterfly attractor was observed in a 10-term system by Pehlivan et al. [24]. Bao et al. found chaos in a nine-term system with four line equilibria [25]. By using three quadratic nonlinearities, Vaidyanathan constructed an eight-term polynomial chaotic system [26]. A seven-term chaotic system with a single cubic nonlinearity was reported in [27]. A Lorenz system family with six terms was presented by Pehlivan and Uyaroglu [28]. However, there were few chaotic systems with five terms [29,30]. Five-term systems are attractive because of their algebraic simplicity. Especially, a fiveterm system is one of simple continuous systems which generate chaos [31]. Thus, the purpose of this work is to study a five-term system with hyperbolic sine nonlinearity and its chaos anti-synchronization. Moreover, such a five-term system is a system without equilibrium [32][33][34][35][36].

The system with five terms and its chaos
A five-term system is studied in this work. The simple system with five-term is described bẏ in which a and b are positive parameters (a,b > 0). Specially, there is a hyperbolic sine term in system (1). We set the right-hand side of system (1) to find its equilibrium points: By comparing Equations (2) and (4), we confirm that system (1) is a non-equilibrium one due to b > 0. When a = 1 and b = 2, system (1) displays chaotic behaviour for initial conditions (x(0), y(0), z(0)) = (0, 0.1, 0) as shown in Figure 1. Chaos in system (1) is also verified by the positive Lyapunov exponent L 1 = 0.0977. It is noting that few chaotic systems without equilibrium have been found [32][33][34][35][36].
Multistability leads to complex behaviours in a dynamical system [37][38][39]. Multistability features have been investigated in numerous systems recently [40][41][42][43]. Interestingly, we have found that system without equilibrium (1) displays various behaviours when changing the initial conditions. Figure 2 illustrates the multistability property of the five-term system. In addition, the bifurcation diagram versus initial conditions is reported in Figure 3. We set x(0) = 0 and z(0) = 0 while y(0) is changed from 0.5 to 3.5.

Electronic circuit of the five-term system
The dynamics of the simple five-term no-equilibrium system has been investigated in the preceding section by using numerical methods. It is revealed that system (1) exhibits complex dynamical behaviours including chaos and multistability. In this section, we design and implement an electronic circuit in PSpice capable to mimic the dynamics of system (1) in order to validate the numerical results carried out previously. The schematic diagram of the proposed electronic circuit for system (1) is depicted in Figure 4.
The circuit comprises seven resistors, three capacitors, two analogue multipliers chips (AD633JN), a pair of semiconductor diodes (1N4148) and a quadruple operational amplifier (TL084CN). The analogue multipliers and pair of semiconductor diodes connected in anti-parallel are used to implement respectively, the quadratic nonlinearity and the hyperbolic sine term. The current-voltage characteristic (I−V ) of the pair of semiconductor diodes (D1 and D2) is defined by the following Shockley diode equation [44]: where I s is the saturation current of the junction, η is an ideality factor (1 < η < 2) and V T is a thermal voltage. By applying Kirchhoff 's laws into the circuit of Figure 4, we obtain its mathematical model given by the following set of three coupled first-order differential equations: where V x , V y and V z are the output voltages of the operational amplifiers OP_1, OP_2 and OP_3, respectively. System (6) is equivalent to system (1) with the following settings of variables and parameters:  Figure 5.
One can see from Figure 5 that the PSpice results agree with those obtained numerically. These results confirm that the proposed electronic circuit is capable to mimic the dynamical behaviours of system (1).

Anti-synchronization of the five-term system
Investigation of chaos synchronization is vital in the theoretical issues and engineering applications [45][46][47][48]. Authors proposed numerous kinds of synchronization such as complete synchronization, phase synchronization, anti-phase synchronization, lag synchronization, anticipating synchronization, projective synchronization and anti-synchronization [49][50][51][52][53][54][55][56]. It is noted that anti-synchronization is an attractive scheme, where two dynamical systems are synchronized in amplitude, but with opposite sign [57][58][59]. Anti-synchronization was applied in different areas, for example temporal pattern recognition [60], memristive neural network [61], multi-degree-of-freedom dynamical system [62], security communication [63] and coupled systems [64,65]. Motivated by the fact that chaos anti-synchronization phenomena are of fundamental importance in the study of dynamical systems, in this section, the antisynchronization of two non-equilibrium systems (master and slave ones) is reported. The master system with five terms is described bẏ In system (7), a and b are unknown parameters. By using the adaptive control u = [u x , u y , u z ] T , the slave  system with five terms iṡ We calculate the state errors of system (7) and system (8): It is noted thatâ andb are the estimations of the unknown parameters (a, b), thus we define the parameter estimation error: e a = a −â, For getting the anti-synchronization (x 2 = −x 1 , y 2 = −y 1 , z 2 = −z 1 ), we introduce the following adaptive control: with positive gain constants (k x > 0, k y > 0, k z > 0). Moreover, we construct the parameter update law: We can confirm the anti-synchronization when applying adaptive control law (11) and parameter update law (12) as follows.
From (9), we geṫ e x = −k x e x , e y = −e a (sinh (x 1 ) + sinh (x 2 )) − k y e y , A simple calculation of (10) giveṡ e a = −ȧ, Figure 6. Time history of anti-synchronization errors between master five-term system (7) and slave five-term system (8). By combining (15), (16) and (14), it is simple to calculate the differentiation of the Lyapunov function: As a result, the anti-synchronization is obtained due to e x → 0, e y → 0, and e z → 0 exponentially as t → ∞ [66]. For the numerical example, we fixed the parameter values We assume that the initial states of master system (7), slave system (8) and the parameter estimate are given by For selected gain constants k x = 4, k y = 4, and k z = 4, the time-history of the anti-synchronization errors e x , e y , e z is reported in Figure 6. In addition, the time series of state variables of the master and slave systems are displayed in Figure 7. We observe the time evolution in opposition of states variables of the master and slave systems which is the signature of anti-synchronization process.

Conclusions
This work has introduced an attractive chaotic system with five-terms, which include a hyperbolic sine term.
The simple system has no equilibrium and displays different behaviours depending on initial conditions. Chaotic behaviour of the system is validated by a circuit, in which the hyperbolic sine term was realized with two diodes. Anti-synchronization of the system has been obtained by designing an adaptive control and illustrated by a numerical example. In our future works, applications of such a system with five terms will be investigated.