Memristor chaotic systems usually have extremely rich dynamical behaviors. In the subsection, the extreme multiple steady states and the state switching will be analyzed.
4.1 Extreme multistability
Extreme multistability is when system states frequently switch between different steady states as the change of initial conditions. Numerical experiments uncover that the system has extreme multistability when the values of parameters are fixed in Table 2.
Table. 2 Parameter values of memristor chaotic systems for extreme multistability |
Parameters | α | β | γ | D | E | F | R | K |
Values | 10 | 6 | 1 | 1.4 | 0.6 | −0.2 | 0 | −2 |
Let the initial conditions of the system be (x(0), 0.1, 0.1, 0.1), and x(0) changes from − 10 to 10, the bifurcation diagram and Lyapunov exponents of the system are obtained, as shown in Fig. 8. It can be seen from Fig. 8(a) that the system has rich dynamics. To be specific, chaos and quasi-periodic motion emerge alternately with the increase of component x(0) from − 10 to 10, and the amplitude of quasi-periodic motion varies with the change of component x(0). In particular, there are two Lyapunov exponents greater than zero in some intervals, as shown in Fig. 8(b). These mean that the system has an infinite number of coexisting attractors, including hyperchaotic attractors and quasi-periodic attractors. Some typical singular attractors are explored further below.
Let values of x(0) be 4.028, 6.07, and 9.198, phase diagrams of typical quasi-periodic attractors of the system in the x-z plane are displayed, as shown in Figs. 9(a), 9(c), and 9(e), which exhibit the variety of topology of system trajectories. And the corresponding Poincare sections of the system in the x-z plane are also given, as shown in Figs. 9(b), 9(d), and 9(f), confirming that three quasi-periodic attractors have diverse topological structures.
Similarly, the values of component x(0) are selected as − 2.026, 6.273, and 8.397, and the phase trajectory diagrams of typical chaotic attractors of the system in the x-y plane are given, as shown in Figs. 10(a), 10(c), and 10(e), respectively. Because these attractors are similar, it is not easy to differentiate whether they are the same attractor. The Poincare sections of the corresponding attractors are obtained, as shown in Figs. 10(b), 10(d), and 10(f). They confirm that the three types of attractors are different. In short, the system has various switching of the hyperchaos and quasi-periodic states with the perturbation of component x(0).
In order to explore the dynamics with the disturbance of components y(0), z(0) and w(0), initial conditions are set as {x(0), y(0), 0.1, 0.1} and the dynamics map of the system with respect to x(0) and y(0) is given, as shown in Fig. 11(a). It can be seen from Fig. 11(a) that the maximal Lyapunov exponent of the blue domain is approximately zero, the system is quasi-periodic; the maximal Lyapunov exponent of the green domain is approximately 0.4, the system is chaotic or hyperchaotic. The crisscross of various coloration domains in the dynamics map means that the system state switches between quasi-periodic state, chaos, or hyperchaos with the perturbation of components x(0) and y(0). There is, in addition, the other dynamics map with respect to z(0) and w(0), is given, as shown in Fig. 11(b), with initial conditions {0.1, 0.1, z(0), w(0)}. The interleaving of multi-colored regions further confirms that there are multiple coexisting attractors in the system.
In nonlinear dynamical systems, there exist infinite attractors with different topological structures, including periodic, quasi-periodic, chaotic, and hyperchaotic attractors, etc[56–58]. However, the proposed system only presents quasi-periodic, chaotic, and hyperchaotic states. Such dynamic behaviors are rarely reported in relevant pieces of literature.
4.2 Dynamics evolution of the system with the change of parameter
Infinite attractors coexist with different initial conditions and the specific parameter set in the proposed system, as previously discussed. Next, the dynamic evolution of the system is analyzed and discussed with the change of a parameter.
Let parameters (α, γ, R) = (10, 0.5, 0) and initial conditions (x(0), y(0), z(0), w(0)) = (0.1, 0.1, 0.1, 0.1). When parameter β increases monotonically from 0 to 8, the bifurcation diagram of the system is obtained, as shown in Fig. 12(a). The corresponding Lyapunov exponents are calculated, as shown in Fig. 12(b). As can be seen from Fig. 12(a), the steady state of the system converts among the chaos, hyperchaos, and the symmetric quasi-period with the increase of parameter β. And two Lyapunov exponent curves above the horizontal axis are approximatively symmetrical with the other two Lyapunov exponent curves below the horizontal axis (see Fig. 12(b) for details). This indicates that hyperchaos exists in the conservative memristor system. For example, hyperchaos is observed with parameter β = 1.8, as shown in Fig. 13(a), where two maximum Lyapunov exponents are LE1 = 0.177 and LE2 = 0.023. The corresponding Poincare section is obtained, as shown in Fig. 13(b). The existence of random points in the x-z plane confirms the existence of hyperchaos.
When parameter β skips the cut-off point β = 1.857, the system enters the quasi-periodic state. For example β = 1.9, four Lyapunov experiments are approximately zero. The trajectory of the system is depicted in purple, as shown in Fig. 13(c). The Poincare section of the system presents four closed independent rings, as shown in Fig. 13(d). This confirms the existence of the quasi-periodic state.
In brief, the system exhibits a switch between hyperchaos and quasi-period with the change of parameter β. Further numerical simulation shows that the switching is also seen in the other parameter set. This is an uncommon nonlinear phenomenon in memristive chaotic systems.