Coexisting asymmetric behavior and free control in a simple 3-D chaotic system

doi:10.1016/j.aeue.2020.153234

AEU - International Journal of Electronics and Communications

Volume 122, July 2020, 153234

https://doi.org/10.1016/j.aeue.2020.153234 Get rights and content

Highlights

•
A new 3-D asymmetric chaotic system with only one quadratic nonlinearity is proposed.
•
This system shows the bi-stability phenomenon of coexisting asymmetric attractors.
•
The amplitude, frequency, and polarity of the generated signals are freely controlled.
•
The pseudo-multi-scroll attractors are generated from the chaotic system via non-autonomous approach.

Abstract

The design and circuit implementation of flexible chaotic systems with desired properties have been of great interest due to their promising applications in various chaos-based information systems. In this paper, a simple three-dimensional autonomous chaotic system with only one quadratic nonlinearity and six linear terms is proposed. The system is asymmetric, which has two equilibria and can generate a single-scroll attractor. This system exhibits asymmetric bi-stability behaviors of chaotic attractors coexisting with periodic limit cycles and other chaotic attractors. By introducing control functions to coefficients of specific linear term or quadratic term, the system can be extended as an amplitude-controllable, equilibrium-shiftable chaotic system. Consequently, attractor rotation is implemented via polarity symmetry, and pseudo-multi-scroll attractors are further generated via nonautonomous approach. The theoretical analyses are confirmed by both the numerical simulations and circuit verifications.

Introduction

Chaos, a kind of irregular and noise-like behavior, can be generated from simple mathematical models of nonlinear differential equations [1], [2], [3]. The specific properties of chaos, such as unpredictability and high pseudo-randomness, make them widely applied in cryptosystem for secure communications [4], [5], [6]. The design and implementation of chaotic system with desired properties are not only interest in theories but significant in real-world applications. Recently many dynamical systems with various equilibria were proposed, such as plane equilibria [7], [8], [9], curve equilibria [10], [11], [12], [13], stable equilibria [14], [15], [16], no equilibria [17], [18], and multi-character equilibria [19], [20], [21]. Accordingly, they bring many interesting dynamics such as the bi-stability [13], multi-stability [9], [16], [21], [22], [23], [24], and extreme multi-stability phenomena [8], [10], [12], [18], [25]. Besides, complex morphology and topology of the attractors [26], [27], [28], [29], [30], [31], [32] and adjustability of output signals [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], fractional-order chaotic systems [45], and other different characteristics [46], [47] were studied to enhance the practical applications in the filed of nonlinear sciences.

Especially, in order to satisfy the special requirements of the engineering-applicable chaotic signal, the amplitude control (AC), frequency control (FC), and polarity control (PC) are necessary [35], [36], [37], [38], [40], [41]. In 2013, Li and Sprott discussed a general approach for the AC of quadratic systems [35]. This approach is realized by introducing control functions to different coefficients of the quadratic terms. In 2017, Li et al. proposed a free-controlled chaotic oscillation in which all terms are quadratic except for a linear function [36]. The AC, FC, and PC can be freely or synthetically realized by changing the linear function. Bao et al. designed a third-order memristive band pass filter chaotic circuit, in which AC can be easily realized by adjusting the potentiometer and the gain of multiplier [37], [38]. Hu et al. developed a switchable chaotic oscillator with independent amplitude-frequency controller [40]. In 2018, Chen et al. proposed a flexible chaotic system, in which the amplitude of variables, Lyapunov exponent, and Kaplan-Yorke dimension were all adjustable [41]. Most recently, Yuan et al. designed a chaotic oscillator with initial-condition-triggered amplitude, frequency and parameter space boosting [42]. Wang et al. reported the attractor rotation phenomenon by changing initial conditions [43]. Sayed et al. realized 2-D rotation of chaotic attractors by using rotation matrix [44].

In the above reviewed systems, multiple nonlinear terms are included in system equations [35], [41], [42], [43], even all terms are quadratic except for a linear term or a constant term [34], [36], [40]. In physical implementations, the quadratic nonlinearity and constant term require additional elements, such as the multiplier and direct-current bias voltage. The algebraic equations’ structure is an index for assessing the simplicity of the system, which means simple implementations in real-world applications [48], [49]. Thus, can these flexible controls be realized in a simple chaotic system with a minimum of nonlinear terms?

In this paper, we propose a new three-dimensional autonomous chaotic system containing only one quadratic nonlinearity and six linear terms. The proposed system is asymmetric with respect to the origin, any principal coordinate axis, even any principal plane, which has advantages in secure communication applications [46]. Although the algebraic structure is simple, the system still has better performance than many others with complex nonlinear terms in term of Lyapunov dimension [47]. Besides, the asymmetry of the system presents the bi-stability phenomenon of coexisting asymmetric attractors. The shiftable equilibrium makes it easy to implement AC, FC, PC, and attractor rotation. When pulse excitations are introduced to the system, pseudo-multi-scroll attractors can be further generated. To the best of our knowledge, this simple chaotic system with so many flexibilities has never been reported.

The remainder of the paper is organized as follows. In Section 2, the model description and basic properties analyses are performed. Dynamical analyses are provided in Section 3. The free control including AC, FC, PC, and attractor rotation is presented in Section 4. The Multisim circuit simulations and verifications are provided in Section 5. The last section concludes the paper.

Section snippets

System model

A new 3-D continuous-time system with only one quadratic nonlinearity is proposed as follows $\{\begin{matrix} \dot{x} = - x + y, \\ \dot{y} = - ay + bz - xy, \\ \dot{z} = - y + z, \end{matrix}$ where a and b are two system parameters. The new system doesn’t exhibit any symmetry, i.e., system (1) is asymmetric, and similar with the Rössler system (case 3) [3]. A comprehensive study of the system is easily performed because entire two-parameter space can be explored and all types of dynamical behaviors and the bifurcation boundaries can be identified [39].

Dissipation and existence of attractor

The general

Lyapunov exponent and Kaplan-Yorke dimension

For the autonomous dynamical system, the positive largest Lyapunov exponent means chaos. When $a = 0.2$ and $b = 5$ , system (1) is chaotic. The finite-time Lyapunov exponents calculated by Wolf’s method [50] are $L_{1} = 0.183, L_{2} \approx 0$ , and $L_{3} = - 0.938$ . Another indicator of the complexity degree of dynamical system is local Kaplan-Yorke dimension ( $D_{KY}$ ), which is defined by $D_{KY} = D + \frac{1}{| L_{D + 1} |} \sum_{j = 1}^{D} L_{j},$ where D is the largest integer satisfying $\sum_{j = 1}^{D} L_{j} ⩾ 0$ and $\sum_{j = 1}^{D + 1} L_{j} < 0, L_{j}$ ( $j = 1, 2, 3$ ) are three Lyapunov exponents that sorted

Free control of the chaotic system

By introducing control functions to the coefficients of specific linear term or quadratic term of system (1), the amplitude, frequency, and polarity of the generated signals are freely controlled without using amplifiers. In the following analyses, parameters $a = 0.2$ and $b = 5$ are fixed, and only the control functions are used to realize AC, FC, PC, etc.

Circuit implementation of the chaotic system

In this section, the proposed system is implemented in NI Multisim 12.0 circuit simulation software, as shown in Fig. 13. The op-amps, resistors, and capacitors are used to implement the modules of integration, addition, and subtraction of the state variables $x, y$ , and z, respectively. The multiplication is implemented by using analog multipliers M, M1, and M2, and the switching sequences are generated from the function generators XFG1 and XFG2. Besides, the single pole double throw (SPDT) S1

Conclusion

In this paper, we proposed a novel three-dimensional chaotic system with only one quadratic nonlinearity and six linear terms. The system has one fixed zero equilibrium and another shiftable none-zero one. Although the algebraic equations are simple, the system still has complex dynamical behaviors. By turning parameters and initial conditions, coexisting bifurcations and coexisting asymmetric attractors were investigated by using Lyapunov exponent, bifurcation diagrams, phase portraits, and

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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^☆: This work was supported by the National Natural Science Foundations of China under Grant No. 61473202.

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Regular paperCoexisting asymmetric behavior and free control in a simple 3-D chaotic system☆

Highlights

Abstract

Introduction

Section snippets

System model

Dissipation and existence of attractor

Lyapunov exponent and Kaplan-Yorke dimension

Free control of the chaotic system

Circuit implementation of the chaotic system

Conclusion

Declaration of Competing Interest

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Regular paper
Coexisting asymmetric behavior and free control in a simple 3-D chaotic system☆