Abundant Coexisting Multiple Attractors’ Behaviors in Three-Dimensional Sine Chaotic System

Wu, Huagan; Bao, Han; Xu, Quan; Chen, Mo

doi:https://doi.org/10.1155/2019/3687635

Complexity

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Abstract Introduction System Model Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability

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Research Article | Open Access

Volume 2019 | Article ID 3687635 | https://doi.org/10.1155/2019/3687635

Abundant Coexisting Multiple Attractors’ Behaviors in Three-Dimensional Sine Chaotic System

Huagan Wu,¹Han Bao,²Quan Xu,¹and Mo Chen¹

Guest Editor: Viet-Thanh Pham

Received15 Oct 2019

Revised02 Nov 2019

Accepted15 Nov 2019

Published06 Dec 2019

Abstract

This paper presents a novel and simple three-dimensional (3-D) chaotic system by introducing two sine nonlinearities into a simple 3-D linear dynamical system. The presented sine system possesses nine equilibrium points consisting of five index-2 saddle foci and four index-1 saddle foci which allow the coexistence of various types of disconnected attractors, also known as multistability. The coexisting multiple attractors are depicted by the phase plots and attraction basins. Coexisting bifurcation modes triggered by different initial values are numerically simulated by two-dimensional bifurcation and complexity plots under two sets of initial values and one-dimensional bifurcation plots under three sets of initial values, which demonstrate that the abundant coexisting multiple attractors’ behaviors in the presented sine system are related not only to the system parameters but also to the initial values. A simulation-oriented circuit model is synthesized, and PSIM (power simulation) screen captures well validate the numerical simulations.

1. Introduction

Recently, numerous nonlinear dynamical systems have been reported that they present the coexistence of two or more disconnected attractors with their isolated attraction basins. The coexisting phenomena of two or more attractors have been encountered in nonlinear oscillating circuits [1–5], biological neuron models [6, 7], Hopfield neural networks [8–11], vibroimpact system [12], superconducting quantum interference device oscillators [13], pure mathematical systems [14–17], and so on. This striking phenomenon, also known as multistability, demonstrates that the system initial values do play an important role in the emergence of complex coexisting attractors’ behaviors [18, 19]. For a multistable dynamical system, it is usually challenging to predict the final steady state to which the dynamical system will tend for a given initial value since a small disturbance in the initial value can alter the steady state of such dynamical systems [20–23]. Multistability has great application potentials in the chaos-based secure communication and information encryption [24–27], but efficient prediction and control methods should be employed to make these dynamical systems in the desired oscillating modes [28–32].

Usually, an effective method for implementing the initial-related multistability is to lead one, two, or more generic or extended memristors in various existing circuits and systems [15–17, 33–36]. Memristor-based circuits and systems with different types of equilibrium points are easy to exhibit coexisting attractors’ behaviors of multistability. Comparatively speaking, another beneficial and simple method for generating initial value offset-boosted coexisting attractors is to put periodic trigonometric functions into specific offset-boostable dynamical systems [37–41]. When the cyclic periods for the periodic functions are identical, any attractor will be copied by periodic offset boosting the initial values [37]. However, due to the reported offset-boostable dynamical systems with self-contained nonlinearities, the newly constructing multistable dynamical systems become relatively complicated [38–41], not convenient for theoretical analyses and hardware circuit implementations. The algebraic simplicity of system’s structure and topological complexity of chaotic attractors are benefits for developing chaos-based cryptosystems [42]. In this paper, based on a simple 3-D linear dynamical system and two newly introduced sine nonlinearities, a novel and extremely simple 3-D sine chaotic system is readily constructed, from which abundant coexisting multiple attractors’ behaviors are observed [43].

The rest is organized as follows. In Section 2, a novel and simple 3-D sine chaotic system is presented. It has nine equilibrium points consisting of five index-2 saddle foci and four index-1 saddle foci, resulting in the coexistence of up to six types of disconnected attractors. In Section 3, by two-dimensional bifurcation and complexity plots under two sets of initial values and one-dimensional bifurcation plots under three sets of initial values, coexisting bifurcation modes are numerically simulated to demonstrate the abundant coexisting multiple attractors’ behaviors. In Section 4, with the simulation-oriented circuit model, PSIM screen captures validate the numerical simulations. The conclusion is summarized in Section 5.

2. System Model and Its Coexisting Multiple Attractors

By introducing two sine nonlinearities with two coupling coefficients into a simple 3-D linear dynamical system, a novel 3-D sine chaotic system with simple algebraic equations is easily achieved, which is modeled bywhere x, y, and z are the three state variables and k₁ and k₂ are the two positive constants.

The presented sine system in (1) is symmetric about the origin and dissipative. The symmetric property can be demonstrated by the invariance of system (1) with respect to the transformation . The dissipativity is explained by

Thus, the orbits are finally confined to a specific subset with zero volume and its asymptotic motion settles onto a standalone attractor.

The equilibrium points of the presented sine system in (1) are obtained by solving the following equations:which is expressed as

The values δ and σ can be yielded by solving the following transcendental functions:respectively.

The Jacobian matrix at E is given asso that the characteristic polynomial is derived aswhere

The above characteristic polynomial implies that Jacobian matrix (7) has three nonzero roots. For these roots, Routh–Hurwitz conditions are given asi.e.,

If the conditions in (11) are satisfied, i.e., and , E is stable, leading to the existence of the point attractor. Otherwise, if any one of the conditions in (11) is not satisfied, i.e., or , E is unstable, resulting in that unstable behaviors may be triggered in the presented sine system.

Denote k₁ = k₂ = k and take k = 3.6 and 5 as two examples. The values δ and σ of the equilibrium point E in (4) are the intersection points of two function curves h₁ and h₂ described by (5) and (6), as shown in Figures 1(a) and 1(b), respectively, from which nine pairs of δ and σ are obtained by inspecting the intersection points, indicating the existence of nine equilibrium points in the presented sine system. With these equilibrium points, the three nonzero eigenvalues are calculated from the Jacobian matrix in (7) and the corresponding stabilities can be determined, as listed in Table 1. The calculation results illustrate that five index-2 saddle foci (Index-2 USF, for short) and four index-1 saddle foci (Index-1 USF, for short) can be found, which could emerge five disconnected attracting regions when these attracting regions cannot be linked with each other, leading to the coexistence of disconnected attractors.

(a)

(b)

For k = 3.6, 5, and several sets of the initial values (labeled in Figure 2), the phase plots of coexisting multiple attractors projected on the x-y plane are depicted in Figures 2(a) and 2(b), respectively. In Figure 2(a), a chaotic attractor coexisted with twin small-size period-1 limit cycles, twin large-size period-1 limit cycles, and a relatively larger size period-1 limit cycle. By contrast, in Figure 2(b), two chaotic attractors coexisted with twin large-size period-1 limit cycles and a relatively larger size period-1 limit cycle. Therefore, up to six types of coexisting multiple attractors are numerically disclosed in the 3-D sine chaotic system because of the attracting and repelling interactions between the five index-2 saddle foci and four index-1 saddle foci.

(a)

(b)

To detect the attracting regions of the coexisting multiple attractors given in Figure 2, the attraction basins in the initial value plane are used to classify different dynamical behaviors [44]. For the two coupling coefficients used in Figure 2, the attraction basins in the - plane with = 10⁻⁶ are depicted in Figure 3. The attracting regions painted by different colors represent the initial value regions corresponding to different long-term oscillating states, i.e., coexisting multistable states, which are identical with the colored trajectories appearing in Figure 2. Thus, the attraction basins show the relatively complicated manifold structures along with basin boundaries. Meanwhile, the numerical results in Figure 3 demonstrate the emergence of multistability in the presented sine system.

(a)

(b)

It can be concluded that due to the appearance of five index-2 saddle foci and four index-1 saddle foci and their interactions with each other, some disconnected attracting regions are thereby formed in the neighborhoods around these unstable saddle foci, resulting in the generation of coexisting multiple disconnected attractors.

Because the presented sine system in (1) is symmetric about the origin, the disconnected attracting regions have locally symmetric behaviors, which are well exhibited in Figures 2 and 3. For this reason and convenient analysis, three sets of the initial values (10⁻⁶, 0, 0), (10⁻⁶, 3, 0), and (10⁻⁶, –3, 0) are considered in next numerical plots. The fourth-order Runge–Kutta algorithm with the 0.01 s time step and (700 s, 800 s) time interval is used for depicting the phase plots and bifurcation plots, whereas the fourth-order Runge–Kutta-based Wolf’s method with the 0.01 s time step and 20 ks time end is adopted for calculating the Lyapunov exponents.

Firstly, two-dimensional bifurcation plots (bifurcation diagrams and dynamical maps) [23] are employed to show complex dynamical behaviors in the presented sine system intuitively, as shown in Figures 4 and 5. Here, both the coupling coefficients k₁ and k₂ are simultaneously increased in the region [2, 8] and two sets of initial values (10⁻⁶, 0, 0) and (10⁻⁶, 3, 0) are chosen. Note that the exhibited two-dimensional bifurcation behaviors are similar to each other for the initial values (10⁻⁶, 3, 0) and (10⁻⁶, –3, 0) due to the system symmetry.

(a)

(b)

(a)

(b)

As shown in Figure 4, the two-dimensional bifurcation diagrams in the k₁-k₂ parameter plane are obtained by calculating the periodicities of state variable x, which demonstrate rich and complex coexisting dynamical behaviors related to system parameters and initial values. The stable points and chaotic attractors distribute in the black- and red-colored regions, respectively, and the periodic attractors with different periodicities situate in the other colored regions. Comparing Figure 4(b) with Figure 4(a), there is a big difference between the two dynamical behaviors in the lower right regions, which is triggered by the initial values-dependent multistability in the presented sine system, leading to the coexistence of multiple bifurcation modes.

As shown in Figure 5, the two-dimensional dynamical maps in the k₁-k₂ parameter plane under two sets of initial values are depicted by evaluating the values of the largest Lyapunov exponent. The yellow-red-white colored regions with different positive values of the largest Lyapunov exponent represent different chaotic behaviors, the black-colored regions with different negative values of the largest Lyapunov exponent only stand for stable point behaviors, and the black-yellow colored regions with the zero largest Lyapunov exponent represent different periodic behaviors. In a similar manner, the dynamical behaviors described by the dynamical maps in Figures 5(a) and 5(b) are of great difference, which manifest how coexisting dynamical behaviors evolve for different initial values.

Similarly, the two-dimensional spectral entropy-based complexity plots in the k₁-k₂ parameter plane are displayed in Figure 6, where two sets of initial values (10⁻⁶, 0, 0) and (10⁻⁶, 3, 0) are employed. On the basis of the Fourier transform [32, 45], the complexity values are obtained by calculating the spectral entropy of the time sequence of the variable x. The relatively large complexity value in Figure 6 indicates the appearance of an irregularly chaotic sequence, whereas the relatively small complexity value in Figure 6 represents the occurrence of a regularly periodic sequence. For the two sets of different initial values, there are some differences in the complexity plots between Figures 6(a) and 6(b), implying that the system initial values have great effects on the dynamical behaviors of the presented sine system.

(a)

(b)

Therefore, the dynamical maps shown in Figure 5 and complexity plots shown in Figure 6 can reflect the dynamical evolutions with the variations of the system parameters and initial values, which are the effective supplements to confirm the coexisting dynamical behaviors depicted by the bifurcation diagrams in Figure 4.

To visualize the coexisting multiple bifurcation modes related to the initial values, three sets of initial values (10⁻⁶, 0, 0), (10⁻⁶, 3, 0), and (10⁻⁶, –3, 0) are considered and both the coupling coefficients k₁ and k₂ are simultaneously increased in the region [2, 8]. Denote k₁ = k₂ = k as a bifurcation parameter. The one-dimensional bifurcation plots with the variation of the system parameter k are shown in Figure 7. In Figure 7(a), the bifurcation diagrams drawn by the black, blue, and red trajectories correspond to those initiated from the initial values (10⁻⁶, 0, 0), (10⁻⁶, 3, 0), and (10⁻⁶, –3, 0), respectively. And in Figure 7(b), the first two Lyapunov exponents associated with three sets of initial values are drafted in the upper, middle, and bottom of Figure 7(b), which entirely match with the bifurcation diagrams in Figure 7(a). Therefore, when more initial values are considered, more complicated coexisting multiple bifurcation modes can be revealed in the presented sine system.

(a)

(b)

Observed from Figure 7, abundant coexisting multiple attractors’ behaviors related to the initial values are exhibited, including stable points, periodic oscillations, and chaotic oscillations along with period-doubling bifurcations, tangent bifurcations, and crisis scenarios. When two sets of initial values (10⁻⁶, 3, 0) and (10⁻⁶, –3, 0) are chosen, both the depicted dynamical behaviors in Figure 7 are basically identical over the entire parameter region, with only slight differences in the parameter region (6.94, 7.58). However, when the other two sets of initial values (10⁻⁶, 0, 0) and (10⁻⁶, 3, 0) are chosen, both the depicted dynamical behaviors in Figure 7 have big differences in the parameter region (2.96, 5.24). As the parameter k is increased in this parameter region, the moving orbit for (10⁻⁶, 0, 0) goes into chaotic oscillating state at k = 3.01 via period-doubling bifurcation route and mutates into periodic oscillating state at k = 3.76 via chaos crisis, whereas the moving orbit for (10⁻⁶, 3, 0) turns into periodic oscillating state from stable resting state at k = 3.14 and enters into chaotic oscillating state at k = 4.34 via period-doubling bifurcation route with two relatively larger periodic windows. Of course, in the parameter region (6.94, 7.58), some slight differences between the depicted dynamical behaviors under two sets of initial values (10⁻⁶, 0, 0) and (10⁻⁶, 3, 0) can be seen for the presented sine system as well.

Except for the two examples in Figure 2, other examples to exhibit coexisting multiple attractors’ behaviors are given in Figure 8, where four sets of phase plots in the x-y plane are provided together for the initial values (10⁻⁶, 0, 0), (10⁻⁶, 3, 0), and (10⁻⁶, –3, 0). When k = 3, the coexistence of a period-4 limit cycle and a pair of symmetric points is exhibited in Figure 8(a). When k = 4.5, the coexistence of a large size period-1 limit cycle and a pair of symmetric period-3 limit cycles is demonstrated in Figure 8(b). When k = 7, the coexistence of a chaotic attractor and a period-5 limit cycle is illustrated in Figure 8(c). However, when k = 8, the coexistence of two chaotic attractors with different topologies is disclosed in Figure 8(d). Consequently, various types of coexisting attractors’ behaviors can be found in the presented sine system.

(a)

(b)

(c)

(d)

4. Validations by the Simulation-Oriented Circuit Model

By employing PSIM Version 9.0.3 software, the simulation-oriented circuit model for implementation of the presented sine system is synthesized and its screen shot is given in Figure 9, in which three operation channels containing three integrators, three inverters and two sine function converters are used to implement three state variables x, y, and z, respectively.

Based on the simulation-oriented circuit model shown in Figure 9, the state equations for the capacitor voltages , , and are described bywhere C₁ = C₂ = C₃ = C, R_k1 = R/k₁, and R_k2 = R/k₂. When RC = 10 kΩ × 10 nF = 100 μs, i.e., R = 10 kΩ and C = 10 nF, the circuit parameters R_k1 and R_k2 for PSIM circuit simulations can be conveniently determined.

According to the system parameters k₁ and k₂ and the initial values used in Figure 2, the circuit parameters R_k1 and R_k2 have the same values, i.e., R_k1 = R_k2. When R_k1 = R_k2 = 2.78 kΩ and 2 kΩ, respectively, PSIM screen captures are obtained in Figure 10, where the initial voltages and of the capacitors C₁ and C₃ are always fixed as 1 μV and 0 V, respectively, and only the initial voltage of the capacitor C₂ is adjusted as different initial values.

(a)

(b)

Similarly, based on the system parameters k₁ and k₂ and three sets of initial values used in Figure 8, the circuit parameters are selected as R_k1 = R_k2 = 3.33 kΩ, 2.22 kΩ, 1.43 kΩ, and 1.25 kΩ, respectively. The corresponding PSIM screen captures are attached in Figure 11, where the initial voltages , , and of the capacitors C₁, C₂, and C₃ are assigned as = 1 μV, = 3 V (or 0 V and –3 V), and = 0 V, respectively.

(a)

(b)

(c)

(d)

Figure 11

PSIM screen captures of coexisting attractors in the plane for different values of R_k1 and R_k2. (a) Period-4 limit cycle coexisted with a pair of symmetric points at R_k1 = R_k2 = 3.33 kΩ. (b) Large-size period-1 limit cycle coexisted with a pair of symmetric period-3 limit cycles at R_k1 = R_k2 = 2.22 kΩ. (c) Chaotic attractor coexisted with period-5 limit cycle at R_k1 = R_k2 = 1.43 kΩ. (d) Coexisting chaotic attractors with two topologies at R_k1 = R_k2 = 1.25 kΩ.

PSIM circuit simulations in Figure 11(d) are slightly different from MATLAB numerical simulations in Figure 8(d), which are mainly caused by the inconsistently transient behaviors due to the existence of simulation errors [46]. Ignoring the tiny differences between MATLAB numerical simulations and PSIM circuit simulations, the results in Figures 10 and 11 effectively validate the coexisting attractors’ behaviors disclosed in Figures 2 and 8.

Besides, it should be mentioned that the sine function terms are the two key units for realizing the proposed 3-D sine chaotic system. In the analog circuit experiments [47], the sine function terms can be physically implemented using two AD639AD trigonometric function converters. But the system initials, corresponding to the initial capacitor voltages, are hardly set in the experimental measurements. In contrast, in the digital circuit experiments [48], the sine function terms can be directly achieved by calling IP cores in CORDIC library of FPGA and the system initials can be readily preset. Therefore, a feasible way to realize the proposed 3-D sine chaotic system could be implemented on the FPGA, which is addressed in our future paper.

5. Conclusion

The autonomous chaotic systems can generate the conventional self-excited attractors as their oscillations are excited from the unstable determined equilibrium points. The mechanism for constructing chaotic systems with coexisting multiple attractors is based on the fact that the system equilibrium points can be reinstalled by newly introduced sine nonlinearities, leading to the great variations of their number, characteristics, and distributions [49]. Therefore, by introducing two sine nonlinearities into a simple 3-D linear dynamical system, this paper presented a novel and simple 3-D sine chaotic system with the reinstalled five index-2 saddle foci and four index-1 saddle foci, from which the abundant coexisting multiple attractors’ behaviors were thereby revealed by numerical simulations, such as phase plots, attraction basins, two-dimensional bifurcation and complexity plots, and one-dimensional bifurcation plots, and finally validated by PSIM circuit simulations. The algebraic simplicity of system structure and topological complexity of chaotic attractor are a long-term goal for seeking a new chaotic system with coexisting behaviors, which could acquire wide interest for its chaos-based engineering applications [42, 50].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the grants from the National Natural Science Foundations of China under Grant nos. 51607013, 61601062, and 61801054 and Natural Science Foundation of Jiangsu Province, China, under Grant no. BK20191451.

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Copyright © 2019 Huagan Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Complexity

Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability

Abundant Coexisting Multiple Attractors’ Behaviors in Three-Dimensional Sine Chaotic System

Abstract

1. Introduction

2. System Model and Its Coexisting Multiple Attractors

3. Initial Values-Related Coexisting Multiple Bifurcation Modes

4. Validations by the Simulation-Oriented Circuit Model

5. Conclusion

Data Availability

Conflicts of Interest

Acknowledgments

References

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