Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability

Abstract

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

“…Multistability refers to the phenomenon that the system shows different dynamic characteristics and different attractors coexist under same parameters [62]. In recent years, the study of multistability and coexistence attractors is a hot topic in nonlinear dynamics [63][64][65][66][67][68][69][70]. Lai et al [63] showed the coexistence behavior of different attractors under different initial conditions and parameter values, such as four limit cycles, and two double-scroll attractors with a limit cycle.…”

“…is new system had no equilibrium point, but it could also show rich and complex hidden dynamics. Zhang et al [66] introduced a state variable into a 3D chaotic system and then analyzed the dynamic characteristics of the new system under different initial conditions, proving that the new system has extreme multistability. In fact, various systems exhibiting multistability have been proposed.…”

A Novel Chaotic System with Two Circles of Equilibrium Points: Multistability, Electronic Circuit and FPGA Realization

“…Equation (4) represents two circles of equilibrium points (x 2 + y 2 + x= 0 and x 2 + y 2 −x = 0) touching at the origin in the x-y plane as shown in Figure 1. The bifurcation analysis of a nonlinear dynamical system is useful for knowing the behavior of the system both chaotic or periodic behavior for certain values of the system parameters [18][19][20][21][22][23]. Lyapunov exponent spectrum and bifurcation diagram are obtained for b = 4.5, c = 1 as a varies between 4 to 15 and initial condition X(0) = (0.01, 0.02, 0.01).…”

A Symmetric Controllable Hyperchaotic Hidden Attractor

“…Other many hyperchaotic systems have come out of the Lorenz-like system [28][29][30][31]. Some other hyperchaotic ones have been proposed, including a memristive hyperchaotic system [32,33], fractional order hyperchaotic system [34,35] or hyperchaotic multi-wing system [36,37]. To the best of our knowledge, there is no relevant research on a hyperchaotic hidden attractor with geometric control.…”