Elsevier

Chaos, Solitons & Fractals

Volume 162, September 2022, 112454
Chaos, Solitons & Fractals

A new chaotic system with novel multiple shapes of two-channel attractors

https://doi.org/10.1016/j.chaos.2022.112454Get rights and content

Highlights

  • A chaotic system with novel multiple shapes of two-channel attractors is discovered.

  • By adjusting only one parameter of the system, rich dynamic behavior can be obtained.

  • The system has three different types of multistability phenomena.

  • It has transient behavior and high spectral entropy.

Abstract

In this paper, a three-dimensional nonlinear system with only one equilibrium point is constructed based on the Anishchenko-Astakhov oscillator. The system is analyzed in detail using time-domain waveform plots, phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, basins of attraction, spectral entropy, and C0 complexity (a parameter for dynamic properties). It is found that this system has excellent dynamical behavior: the emergence of novel multiple shapes of two-channel attractors and the gradual evolution of clumped and ring-shaped attractors can be tuned by only one parameter. The system also exhibits multistability with three types of dynamical behavior, namely, coexistence of two types of periodic attractors, and coexistence of quasi-periodic/chaotic attractors at different initial values. Moreover, the system has transient behavior, significantly increasing the complexity of the system. Finally, a hardware circuit mimicking the system is implemented. Such dynamical characteristics can be controlled by only one parameter, which is great cost savings and highly efficient in engineering applications.

Introduction

Chaos is a specific phenomenon of nonlinear systems. Chaotic systems also have a wide range of applications. It has been used not only in the field of electronic information such as neural network [1], [2], image encryption [3], [4], [5], secure communication [6], [7], and integrated chaos generators [8], but also in physics [9], economics [10], and biology [11].

Since the discovery of the first three-dimensional chaotic attractor by Lorenz [12] in 1963, there has been a worldwide motivation for studying chaos. Then, a great of three-dimensional continuous chaotic systems have been discovered, including those due to Rössler [13], Chua [14], Chen [15], Lü [16], and Liu [17]. Based on these classical three-dimensional chaotic systems, some new chaotic systems and hyperchaotic systems are subsequently proposed [18], [19], [20], [21], [22]. With the development of circuit systems, the exploration of chaos theory is no longer limited to the scope of numerical simulation, and the desired chaotic signals can be generated by hardware. Many chaotic circuits have been implemented by using analog circuit arithmetic units composed of nonlinear components [23] to represent chaotic signals at the physical level through circuit signals [24], [25], [26], [27], such as Chua circuit [28] and jerk circuits [29], [30]. The vast majority of the above proposed and other existing chaotic systems can be obtained by adjusting several parameters to obtain different shapes of attractors. However, to the authors' knowledge, chaotic systems that can switch novel many different attractor shapes by adjusting only one parameter have not been proposed. Such a chaotic system has better convenience in specific working scenarios, enabling easy and fast switching between periodic, quasi-periodic, and chaotic states.

Various dynamical characteristics of chaotic systems have also attracted much attention. By adding two perturbation terms (that is, simple constant control parameters) to the Sprott D system and increasing the parameters of the variables, it is possible to change the type and number of equilibrium points and to obtain different shapes of attractors in periodic and chaotic states [31]. By adding a parameter to the quadratic term of the Sprott E system, a generalization based on the original system can be obtained [32]. Similarly, a small perturbation term is added to the first term of the Sprott E system to change the degenerate equilibrium to a stable equilibrium, and by adjusting this control parameter, the system appears with six different shapes of attractors [33]. In three-dimensional autonomous chaotic systems with cubic nonlinear terms, they can converge to the equilibrium point for larger values of the parameters [34]. However, the shapes of the attractors obtained by varying the parameters are similar in these newly proposed systems. By choosing different settings of parameters, the fractional multi-attribute chaotic system (FMACS), can generate some strange attractors with peculiar shapes, which contain self-excited attractors with an unstable equilibrium point and hidden attractors without an equilibrium point [35]. In the recently proposed simple parallel memristor chaotic circuit, 4 parameters affect the system and are capable of generating 11 shapes of chaotic attractors and 4 shapes of periodic attractors [36]. The ability to generate multiple shapes of attractors in a chaotic system is certainly a reflection of its high complexity.

Multistability is a special phenomenon that can be found in many nonlinear systems. It represents the non-uniqueness of the final state of the system depending on the initial conditions under fixed parameters. To a large extent, it gives the system better flexibility and robustness. Under proper control, the system can achieve transitions between different states to adapt to a variety of operating scenarios. A chaotic system with only one equilibrium is described by Sprott et al. [37], which has the coexistence of points, periods, and singular attractors. Li et al. [38] introduced the offset boosting technique to chaotic systems for attractor shifting, and the number of coexisting attractors in that system can be doubled. By generating multiple invariant sets to bulk replicate the attractors of the system in phase space, Lai et al. [39] constructed a system that can generate an infinite number of coexisting attractors, even for an extremely simple chaotic system [40].

Inspired by published research on chaotic systems, this paper introduces a new three-dimensional chaotic system. In the case of changing only one parameter, the system emerges with novel multiple shapes of two-channel attractors and evolves with increasing parameters into clumped attractors without channels and finally into period attractors without channels. Altering only a single system parameter results in 3 types of multistability phenomena and some other dynamical behaviors.

The rest of this paper is organized as follows. Section 2 briefly describes the mathematical model of the new system. Section 3 discusses the dynamical properties of the system. In Section 4, the spectral entropy (SE) and C0 complexity of the system are examined. In Section 5, we give the circuit implementation of the novel chaotic system. Finally, the conclusions of this work are arranged in Section 6 an outlook on the subsequent work is also provided.

Section snippets

System description

The model of chaotic system proposed by Anishchenko and Astakhov [41] can be expressed asẋ=ax+yxzẏ=xż=bz+bIxx2whereIx=0,x01,x>0

With the essential characteristics of chaotic system (nonlinearity and symmetry) absolute value functions are good candidates for the construction of chaotic systems. From a practical point of view, an absolute value term is also easy to implement by ordinary electronic components (diodes, resistors, and operational amplifiers). Moreover, such chaotic systems

Numerical analysis of the proposed system

We give a detailed investigation of the dynamical behaviors of the system (3) numerically. We analyzed the nonlinear characteristics of the proposed system in terms of the Poincaré map, Lyapunov exponent spectra, the bifurcation diagrams, phase diagram, basin of attraction, and SE/C0 complexity, which give us a clearer view of the studied features. Here, all the simulations are performed using the ode45 solver in the MATLAB simulation environment.

Complexity of the system

This section characterized the complexity of the system with SE/C0 complexity by adjusting the value of the parameter c ∈ (0, 40), a = 50, b = 1.7, d = 50, and initial values (1, −1, −2) [51], [52], [53], [54], [55].

Fig. 8 shows the SE and C0 complexity versus the parameter c. It can be observed that after a period of multiplicative period windows, a large continuous chaotic interval appears, followed by a large continuous non-chaotic interval. The above result is consistent with the results of

Circuit implementation

Based on the mathematical model of this system (3), a circuit consisting of AD711JN operational amplifiers, 1 N4007 diodes, AD633JN multipliers, precision potentiometers, resistors, and capacitors was designed, and a voltage of ±12 V was used to supply the operational amplifiers. To ensure that the hardware experimental circuit operates in a suitable dynamic range, the state variables of the system are compressed in a linear proportion as follows:x=uxy=uyz=10uz

As shown in Fig. 9, the circuit

Conclusion

In this paper, based on the Anishchenko-Astakhov oscillator a nonlinear system with only one equilibrium point is proposed by replacing the product term of the piecewise function and the square term in the differential equation of z with the absolute value term |x| and introducing the parameter c in the differential equation of y. The temporal evolution and the mechanism of generation of the two-channel attractor is analyzed in the phase diagram adding a color in the trajectories and also by

Data availability statement

The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding author.

CRediT authorship contribution statement

Chenyang Hu: system analysis, circuit design, and draft writing.

Qiao Wang: system analysis, circuit design, and draft writing.

Zean Tian: supervising the whole analysis and manuscript revision.

Xiefu Zhang: checking the whole analysis.

Xianming Wu: reviewing.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 51660115, U1612442, and 62061008).

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.

References (55)

  • N. Wang et al.

    Hidden attractors and multistability in a modified Chua’s circuit

    Commun Nonlinear Sci NumerSimul

    (2021)
  • Z. Wei

    Dynamical behaviors of a chaotic system with no equilibria

    Phys Lett A

    (2011)
  • X. Wang et al.

    A chaotic system with only one stable equilibrium

    Commun Nonlinear Sci NumerSimul

    (2012)
  • A. Abooee et al.

    Analysis and circuitry realization of a novel three-dimensional chaotic system

    Commun Nonlinear Sci NumerSimul

    (2013)
  • T. Liu et al.

    A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation

    Chaos, Solitons Fractals

    (2021)
  • X. Zhang et al.

    A simple parallel chaotic circuit based on memristor

    Entropy

    (2021)
  • Y. Liu et al.

    The basin of attraction of the liu system

    Commun Nonlinear Sci NumerSimul

    (2011)
  • P.-Y. Xiong et al.

    Spectral entropy analysis and synchronization of a multi-stable fractional-order chaotic system using a novel neural network-based chattering-free sliding mode technique

    Chaos, Solitons Fractals

    (2021)
  • R. Yan et al.

    Multilayer memristive neural network circuit based on online learning for license plate detection

    IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

    (2021)
  • Q. Hong et al.

    Memristive circuit implementation of a self-repairing network based on biological astrocytes in robot application

    IEEE Trans Neural Netw Learn Syst

    (2022)
  • F. Yu et al.

    FPGA implementation and image encryption application of a new PRNG based on a memristive hopfield neural network with a special activation gradient

    Chin Phys B

    (2021)
  • C. Masoller

    Coexistence of attractors in a laser diode with optical feedback from a large external cavity

    Phys Rev A

    (1994)
  • D. Angeli et al.

    Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems

    PNAS

    (2004)
  • E.N. Lorenz

    Deterministic nonperiodic flow

    J Atmos Sci

    (1963)
  • L.O. Chua

    Chua’s circuit 10 years later

    Int J Circ Theor Appl

    (1994)
  • T. Ueta et al.

    Bifurcation analysis of chen’s equation

    Int J Bifurcation Chaos

    (2000)
  • J. et al.

    A new chaotic attractor coined

    Int J Bifurcation Chaos

    (2002)
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    Chenyang Hu and Qiao Wang contributed equally to this work

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