Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points

ABSTRACT We investigate a three-dimensional (3D) robust chaotic system which only holds two non-hyperbolic equilibrium points, and finds the complex dynamical behaviour of position modulation beyond amplitude modulation. To extend the application of this chaotic system, we initiate a novel methodology to construct multiwing chaotic attractors by modifying the position and amplitude parameters. Moreover, the signal amplitude, range and distance of the generated multiwings can be easily adjusted by using the control parameters, which enable us to enhance the potential application in chaotic cryptography and secure communication. The effectiveness of the theoretical analyses is confirmed by numerical simulations. Particularly, the multiwing attractor is physically realized by using DSP (digital signal processor) chip.


Introduction
As an active topic, chaos has been extensively and continuously investigated in classical [1][2][3][4][5][6][7] and quantum fields [8,9] over the last half-century. The study on chaos has well served to promote the exploration of dynamical behaviour, intrinsic structure of the natural system and the design of the new chaotic system, as well as chaos-based application. One of the potential applications of chaos is the image encryption and secure communication based on synchronization technology [10][11][12][13][14][15], since the complicated signal of the chaotic system could be used as a carrier wave or a secure key. It is confirmed that multiwing (or multiscroll) chaotic system can present more complex dynamics than that with few wings (or scrolls) [16].
Since the famous Chua double-scroll circuit was formulated and analysed [17,18], numerous efforts have been exerted to construct and realize multiwing (or multiscroll) chaotic attractors. The basic idea for most of the nonlinear systems to generate multiwing (or multiscroll) attractors is to increase the number of index-2 equilibrium points by introducing multisegment quadratic function [19,20], polynomial function [21], saw-tooth function [22,23], sinusoidal function [24,25], hysteresis function [26,27], stair function [28][29][30], transformation function [31], triangle function [21,32] and others. In addition, there exist many other methods to generate multiwing (or multiscroll) chaotic attractors. For example, Yu reported a class of grid multiscroll chaotic attractors by designing switching linear systems and constructing heteroclinic loops [33,34]. Chen introduced a simple method for constructing multiscroll chaotic systems by selecting a proper unstable linear system and implementing shift transformations [35]. Li and Yu proposed a ringscroll Chua system by introducing a generalized ring transformation [36]. Yu also proposed a general ringmultiscroll system family by introducing a parameterized nth-order polynomial transformation in generalized Lorenz systems [37]. Deng and Lü introduced a switching control scheme for constructing multidirectional multiscroll chaotic attractors in the fractional differential system, which can generate more scroll number than classical model [38]. Tahir presented a class of multiwing attractors with no-equilibrium and with an infinite number of equilibrium points respectively, by using a state feedback controller [39]. Jafari [40], Hu [41], Wang [42] and Escalante-González [43,44] also reported a class of multiscroll system without equilibrium. Bouallegue proposed a family of chaotic attractors with separated multiscrolls by combining the Julia fractal process with Chua's attractor and Lorenz's attractor [45]. Although various design methods emerged, it's still an attractive research interest in seeking new methods for constructing multiwing (or multiscroll) chaotic attractors.
The conception of robust chaos was first introduced by Banerjee when studying the current mode controlled boost converter [46], which is defined by the absence of periodic windows over some range of the parameter space and chaos is the unique state in this range. Banerjee also found the robust chaos in buck converter and proposed the conditions of robust chaos [47]. Subsequently, based on Banerjee's theorem, Min introduced a chaos robustness criterion for a kind of two-dimensional piecewise smooth map (2DPSM) and constructed a 2DPSM with robust chaos feature [48]. Andrecut confirmed robust chaos in a family of onedimensional continuous piecewise smooth maps based on bifurcation structure [49]. These investigations only concern the robust chaos feature over a limited parameter range. Recently, we found that there exists robust chaos in the continuous chaotic system over infinite parameter range, and these parameters can regularly control the signal amplitude (which is called amplitude modulation for the convenience), yet the Lyapunov exponents keep invariable [50][51][52][53][54]. Therefore, it is a promising type of system in the practical application of image encryption, signal processing, synchronization and chaotic communication [53,54]. However, as far as we know, little research on such a system, particularly its extension of application, has been done in the present literatures.
This paper aims at developing a systematic methodology for generating multiwing attractors by constructing a robust chaotic system and modifying the position and amplitude parameters of the system. We first introduce a robust chaotic system and analyse the dynamical properties carefully. It's found that the reported system holds two non-hyperbolic equilibrium points and possesses the feature of position modulation (interpreted as controlling the position of the chaotic signal by system parameter), other than amplitude modulation. Then, with the application of position modulation, we realize the double scheme for wing number in the robust chaotic system by modifying the position parameter. And with the application of amplitude modulation, we further construct multiwing chaotic attractors from the modified system by transforming the amplitude parameter. The proposed design philosophy is different from the existing approach in literatures , and the designing process is concise with only three simple operations. What's more, the signal amplitude, range and distance of the generated wings can be modulated by the control parameters. The reported dynamical behaviours of the multiwing chaotic system are discussed theoretically and confirmed numerically, which displays the potential application in chaotic cryptography and secure communication for the flexible choice of attractor characteristics.
The rest of this paper is outlined as below: in Section2, we introduce a robust chaotic system and discuss the dynamical properties. In Section 3, we give the design scheme and analysis for generating multiwing attractors from the robust chaotic system. In Section 4, we give a DSP-based realization of the multiwing attractor. Finally, the conclusion is drawn in Section 5.

Model description
The considered 3D system is described by are the state variables. The volume contraction of system (1) can be reckoned by the Lie derivative ∇V = ∂ẋ/∂x + ∂ẏ/∂y + ∂ż/∂z = −a, which means that system (1) holds a negative dissipativity when a > 0. Thereby, system (1) is bounded, and the asymptotic motion at time through the flow will settle onto an attractor.
For the positive parameters a, b, c, h, k, we obtain two equilibrium points of system (1), as below Then the corresponding characteristic equation at any When considering the representation of E + or E − , the characteristic equation (2) is simplified as Equation (3) holds one negative real root λ 1 = −a and two pure imaginary roots λ 2,3 = ±i 2ch/a. Therefore, the two equilibrium points are non-hyperbolic, and system (1) doesn't belong to Šil'nikov type chaos. By selecting the parameter set S 0 = {a = 5, b = 1, c = 1, h = 40, k = 1} and the initial condition x(0) = (0.01, 0.01, 0.05), we calculate the three Lyapunov exponents of system (1) as (0.508566, 0, −5.498819) by the orthogonal method [55], confirming the chaotic behaviour. The corresponding chaotic attractors are shown in Figure 1. It can be seen that the chaotic trajectories have two wings alternatively swirling around the two non-hyperbolic equilibrium points.

Dynamics analysis of amplitude and position modulation
Although there are still many chaotic systems of nonhyperbolic type, little attention is paid to the property of robust chaos for this kind of system, see, for instance, Refs [56][57][58] and Refs therein. Recently, we reported a non-hyperbolic system and investigated the dynamics features of robust chaos and amplitude modulation [59]. Zhu investigated the dynamics features of amplitude and position modulations, but the reported chaotic system is hyperbolic with one saddle and two saddle-foci equilibrium points [60]. In this paper, the dynamics feature of the position modulation in the robust non-hyperbolic system is found for the first time, not confined to the amplitude modulation.
Our analysis starts with the linear transformation based on the relation of the system parameter and the mathematical expression of the nonzero equilibrium point [52]. Thus, system (1) is degenerated to the normalized system about parameter k, as below Therefore, the parameter k can modulate the signal amplitude of variables x 1 , x 2 according to 1/ √ k, but the amplitude of signal x 3 keeps constant. In addition, we notice that the characteristic equation (3) is irrespective to parameter k, thus the Lyapunov exponent spectrum remains constant when k varies. The feature of amplitude modulation for system (1) is depicted by means of signal amplitude, Lyapunov exponent spectrum and phase diagram for the initial value x(0) = (0.01, 0.01, 0.05), as shown in Figure 2.
Next, when considering the transformation of Thereby, as the parameter b increases, the position of signal x 3 descends at the speed of 1/c, while the amplitude and position of signals x 1 , x 2 keep invariable. Similarly, we know that characteristic equation (3) is irrespective to the variation of parameter b. Thus, when b changes in a real number field, the Lyapunov exponent spectrum remains invariable. The feature of position modulation for system (1) is illustrated in Figure 3, by means of signal amplitude, Lyapunov exponent spectrum and phase diagram for the initial value x(0) = (0.01, 0.01, 0.05).
It's noticed that for the numerical results depicted by Figures 2 and 3, the Lyapunov exponent spectrums keep fluctuating because of the computational precision and the same initial condition of the system under different parameters k or b. In spite of this, the simulation results agree well with the theoretical analyses. x 1min x 2max x 2min x 3max

Design scheme
First of all, we select the parameter set (1), then the position of signal x 3 ascends about 36 units, compared with the selection of parameter set S 0 , as shown in Figure 4(a). It's known that few trajectories pass through x 3 = 0 plane, but most trajectories stay above x 3 = 0 plane. Secondly, to realize the double scheme for wing number in the robust chaotic system, we start with the chaotic system (1) with the parameter set S 1 , and make the parameter transformation b → bsign(x 3 ) by considering the effect of parameter b on signal position. Then the system of differential equations becomes And the equilibrium points The characteristic equation for all these equilibrium points is expressed as (3) through mathematical derivation, therefore, the four equilibrium points are non-hyperbolic. Yet the equilibrium points for x 3 = 0 play a key role that bridges the positive half-space and the negative half-space with respect to the x 3axis. Thus, a four-wing attractor can be generated from system (6), as illustrated in Figure 4(b). Thirdly, to further construct multiwing chaotic attractors from the modified system (6), we make the parameter transformation k → k(1 − k 1 f (k 2 x 2 )) by considering the effect of parameter k on signal amplitude. Then it yields the following resulting system In system (7), f (·) is a periodic function; k 1 and k 2 are the control parameters, which can control the variation ranges of the signal amplitude and the wing distance, respectively. There exist different choices of the function f (·) for the generation of multiwing attractors. For simplicity, we will focus on the case of f (·) = sin(·) in this paper, though similar results and analysis can be made for other periodic functions. Then system (7) can    be reformulated as With the selection of the parameter set S 1 and k 1 = 3.6, k 2 = 3, the numeral simulations of the attractor projected onto the plane (x 2 , x 3 ) in Figure 5 shows that system (8) holds complicated multiwing attractors. The corresponding Lyapunov exponents are reckoned as 0.594554, 0, −5.574220, which further proves the chaotic character. It is worth mentioning that we approximate the signum function sign(x 3 ) by the hyperbolic function tanh(120x 3 ), as shown in Figure 6, to reduce differential buffeting when calculating Lyapunov exponents [61].
The diagram of bifurcation and the maximal Lyapunov exponent spectrum is computed by varying parameter c from its null value to detect the dynamical transition. As found in Figure 7, the system reaches a chaotic pattern immediately from the lowest value of c. And we notice that periodic windows embedded in the chaotic region are observed as the value of c is increased.

Analysis of attractor
In this subsection, we will consider the influences of parameters k, k 1 and k 2 on the characteristics of the generated wing attractors.
The bifurcation diagram for k ∈ [0, 10] is plotted in Figure 8 to describe the amplitude evolution. Superficially, it seems that the coefficient k can modulate the signal amplitude regularly. But one notices with the careful observation that there emerge visible periodic windows in the chaotic region. In fact, we know that the value of k(1 − k 1 sin(k 2 x 2 )) changes with parameter k. In this sense, one may guess that if the multiwing attractors exist, k is a parameter that directly affects the signal amplitudes of x 1 , x 2 by the power function with the index −1/2, but the amplitude of x 3 keeps invariable. The above analysis can be intuitively confirmed by  numerical results with k = 1 and k = 4, as respectively described in Figure 9. We now analyse the influence of control parameter k 1 on the property of the generated wings. It's obvious that a larger value of k 1 will yield a wider variation range of k(1 − k 1 sin(k 2 x 2 )). As a result, if the multiwing chaotic attractors exist, the variation ranges of the signal amplitude of x 1 , x 2 increase with a larger k 1 . In other words, the range from the innermost wing to the outermost wing in x 1 , x 2 direction increase with a larger k 1 . The analysis is compared in Figure 10 with k 1 = 1 and Figure 11 with k 1 = 4 by numerical simulations.
Finally, we consider the role of k 2 . It's noticed that sin(k 2 x 2 ) is a function with the period as 2π/k 2 , which means that the right-hand side of the third equation of system (8) keeps invariable when x 2 moves to x 2 + 2mπ/k 2 , m ∈ Z. Therefore, if these multiwing chaotic attractors exist, the distance of adjacent wings (called wing distance for the convenience) can be estimated as d = 2π/k 2 . And we further know that a larger k 2 will bring a smaller distance d. In this sense, parameter k 2  can directly affect the wing distance. The analysis can be verified numerically in Figure 5 with k 2 = 3 and Figure 12 with k 2 = 9.

DSP-based realization of the multiwing system
As a microprocessor designed with a special structure, DSP chip holds the features of programmability, flexible interface and high operation speed. Therefore, DSP chip plays an increasingly important role on electronic products research and digital signal processing. In this realization, we adopt TMS320C6747 device, which is a low-power digital signal processor based on a TMS320C674x DSP core with 64-bit running at 375 MHz and floating-point operation, therefore, it is considered to be sufficient for our experiment.
To increase the computational precision of the iterative equation and compress the range of state variables, the variable-scale reduction is first made on system (8).
Considering the condition of a = 5, b = -35, c = 1, h = 40, k = 1, k 1 = 3.6, k 2 = 3, and the compression factors (2,6,25) for variables (x 1 , x 2 , x 3 ), we derive the differential equations system of (8) as And we further execute the time-scale transformation to increase the density of periodic orbits and to capture the wave easily. Subsequently, we obtain the resulted system of (9) with the time-scale factor 10, depicted by Considering that DSP is a digital processor, we discretize the continuous system (10) by the classical Runge-Kutta method with step 0.001. Accordingly, we convert the chaotic digital sequences into analog ones by the multichannel DAC (digital analog converter with the model DAC7724). With the setting of the initial value (0.2, 1.2, 0.5), the oscilloscope observation of the multiwing chaotic attractor is displayed in Figure 13. Compared with the attractors simulated by Matlab in Figure 5, it can be concluded that they have a good qualitative agreement.

Conclusion
Robust chaos is an interesting research field of continuing concern and should be paid closer attention to the dynamics and the application expansion. In this paper, we reported a robust chaotic system with amplitude modulation. And for the first time, we found the properties of position modulation and non-hyperbolic equilibrium points swirled around by wings alternatively in the robust chaotic system. By transforming the forms of position parameter and amplitude parameter, a systematic methodology was presented to construct multiwing attractors from the robust chaotic system. The flexible adjustment of the signal amplitude, range and distance of generated wings by changing the control parameters will add the dynamics complexity of the multiwing system, and consequently will enhance the potential application in chaotic cryptography and secure communication. The non-reported design method of multiwing attractors expanded the application range of robust chaos, and will also consequently enrich the design theory of complicated chaotic system.

Disclosure statement
No potential conflict of interest was reported by the authors.