A new chaotic system with novel multiple shapes of two-channel attractors
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 aswhere
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:
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.
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Chenyang Hu and Qiao Wang contributed equally to this work