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A novel 5-dimensional (5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multi-wing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincaré map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.
It is confirmed that dynamical systems with multi-scroll attractors or multi-wing attractors carry much more complexity than those with fewer attractors, and they have been widely used in some areas, such as image encryption,[1] secure communication,[2,3] and chaotic cryptanalysis.[4] Therefore, during the last few decades the dynamical systems with multi-scroll attractors or multi-wing attractors have attracted a great deal of attention from researchers.[5–7] Meanwhile, various nonlinear control schemes which are based on nonlinear functions are proposed to generate multi-scroll attractors or multi-wing attractors, such as hyperbolic function,[8] trigonometric function,[9,10] and switching controller.[11,12] In addition, the study on realization of multi-scroll attractors or multi-wing attractors makes them observable in analog circuits.[13–16] Particularly, in some fractional order systems,[17,18] and memristor-based systems,[19–22] multi-scroll attractors or multi-wing attractors can also be obtained.
The hidden attractor[23–25] is a hot topic and has attracted lots of interest recently. From a computational point of view, attractors in a chaotic system can be classified as self-excited and hidden attractors. An attractor is called a self-excited attractor if its attraction basin intersects with any neighborhoods of a stationary state (an equilibrium); otherwise, it is called a hidden attractor. The hidden attractors exist in some special dynamical systems, including ones with one stable equilibrium,[26,27] or without equilibrium,[28,29] or with a line equilibrium (infinite equilibria).[30] It is thought that the number of multi-scroll attractors or multi-wing attractors in the dynamical systems has a close relation with the system equilibria, and the dynamical systems with multi-scroll attractors or multi-wing attractors are commonly constructed by extending saddle-focus equilibrium points with index 2. However, for the systems with multi-scroll hidden attractors or multi-wing hidden attractors, the generating mechanism does not make sense because the systems have no equilibrium.
Most of the dynamical systems with hidden attractors only have one or two attractors, and the dynamical systems with multi-scroll hidden attractors or multi-wing hidden attractors are seldom reported.[31–34] In Ref. [31], Tahir et al. constructed a 4D chaotic system which has no equilibrium by applying a state feedback controller to a Lorenz-like system, and the new system has multi-wing hidden attractors. Zhou et al.[32] proposed no equilibrium hyperchaotic multi-wing attractors via introducing a memristor into a multi-wing chaotic system. Moreover, multi-scroll hidden attractors were also observed in an improved Sprott A system.[33,34] Inspired by the above references, we wonder are there some dynamical systems that can present both multi-scroll hidden attractors and multi-wing hidden attractors simultaneously? The answer is yes. In this paper, we propose a novel 5D memristive chaotic system. Compared with the systems in the references,[31–34] the novel chaotic system can present either multi-scroll hidden attractors or multi-wing hidden attractors on different phase planes. As the novel chaotic system has no equilibrium, the number of multi-scroll hidden attractors and multi-wing hidden attractors has nothing to do with the system equilibrium. Particularly, it is sensitive to the transient simulation time and initial values.
The organization of this paper is as follows. In Section
The memristor is a two-terminal electronic element which was predicted by Chua,[35] and it is also recognized as the fourth fundamental electronic element. For a memristor, the magnetic flux φ between the terminals is a function of the electric charge q that passes through the device. In this paper, a flux controlled one is used, and the nonlinear relation between terminal voltage u and terminal current i is defined as follows:
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In this paper, we propose a novel dynamical system whose dynamics is described as follows:
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The parameter k is the key factor that determines the type of solutions.
Case A: when k = 0, the equilibria of Eq. (
Case B: when
According to the definition of hidden attractors, the attractors of the dynamical system under the condition of either case A or case B are hidden. In this work, we focus on the dynamical system under the condition of case B, and the system parameters are chosen as a = 0.25, b = 0.4, c = 2, d = 0.5, e = 0.5, g = 15, h = 0.01, and k = 0.05. Dynamics analyses of the new system are presented in the next subsection.
Obviously, in case B, the new dynamical system has no equilibrium, and any attractors of this system are hidden according to the definition of hidden attractors. When the initial values
According to the chaos theory, Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories of a nonlinear dynamical system in phase space, and one positive Lyapunov exponent indicates that the dynamical system is chaotic. Convergence curves of four largest Lyapunov exponents of system (
Extensive numerical simulations confirm that the number of multi-scroll hidden attractors and multi-butterfly wing hidden attractors of the dynamical system is dependent on the transient simulation time. When the initial values are selected as
The dynamical system generates a different number of multi-scroll hidden attractors and multi-butterfly wing hidden attractors when the transient simulation time changes; this is because the nonlinearity of Eq. (
![]() | Figure 5. (color online) Time series of state variable x with initial values and transient simulation time selected as ![]() |
It is inconvenient to apply this kind of attractor in reality; therefore some schemes should be employed to constrain the nonlinearity of the dynamical system. In Refs. [10] and [33], piecewise sine functions were used to avoid the augment of attractors when the transient simulation time went to infinity. Li et al.[13] handled the problem through synchronizing two systems and obtained the desired number of attractors. In Ref. [14], Ma et al. proposed a simple state feedback controller to constrain the range of state variable x, and the number of multi-scroll attractors could be selected by adjusting two control parameters. In this paper, a state feedback controller which is similar to Ref. [14] is used, and the controlled dynamical system can be rewritten as
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When the initial values
As is known to all, the sensitivity to the initial values is a basic characteristic of chaotic systems. Particularly, it is common that the dynamical systems with hidden attractors have the feature of multistability.[36–38] In this subsection, the effects of the initial values on the dynamics of system (
In Figs.
In conclusion, besides transient simulation time, the initial values can also affect the dynamics of the system. This dynamical system has the feature of multistability, and multiple hidden attractors coexist on the same phase plane.
Circuit implementation is a vital procedure for practical applications in reality, and it is also a feasible way to verify the corrections of numerical simulations and theory analysis. Some fundamental analog devices are used in the realization of controlled dynamical system (
Based on the mathematical model of system (
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The state feedback controller
![]() | Figure 9. (color online) 5 scroll hidden attractors of system (5) on the x–y phase plane obtained by setting voltage sources ![]() ![]() |
A novel 5D chaotic system is proposed, which has no equilibrium or multiple lines equilibria. When certain system parameters are chosen, multi-scroll hidden attractors and multi-butterfly wing hidden attractors can be observed on different phase planes, and the number of multi-scroll hidden attractors and multi-butterfly wing hidden attractors has nothing to do with the system equilibrium. Extensive numerical simulations confirm that both transient simulation time and initial values have a significant impact on the number of hidden attractors. A state feedback controller is designed and added to the first equation of system (
A circuit is designed and implemented in Pspice based on the mathematical model of system (