Coexisting Infinitely Many Nonchaotic Attractors in a Memristive Weight-Based Tabu Learning Neuron

Memristive synaptic weight is a changeable connection synaptic weight. It reflects the self-adaption physical processing in biological neurons. To study its dynamical effect, this paper presents a memristive synaptic weight-based tabu learning neuron model. It is constructed by replacing the resistive self-connection synaptic weight in the tabu learning neuron with a memristive self-connection synaptic weight. The equilibrium point of the memristive tabu learning model is time-varying and switches between no equilibrium state and line equilibrium state with the change of the external current. Particularly, the stability of the line equilibrium state closely relies on the initial state of the memristor, resulting in the emergence of coexisting infinitely many nonchaotic attractors. By employing the bifurcation plots, Lyapunov exponents, and phase plots, this paper numerically reveals the initial state-switched coexisting bifurcation behaviors and initial state-relied extreme multistability, and thereby discloses the coexisting infinitely many nonchaotic attractors composed of mono-periodic, multiperiodic, and quasi-periodic orbits. In addition, PSIM circuit simulations and printed-circuit board-based experiments are executed and the coexisting infinitely many nonchaotic attractors are realized physically. The results well verify the numerical simulations.

A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

This work introduces a new 3-D chaotic system with a line of equilibrium points. We carry out a detailed dynamic analysis of the proposed chaotic system with five nonlinear terms. We show that the chaotic system exhibits multistability with two coexisting chaotic attractors. We apply integral sliding mode control for the complete synchronization of the new chaotic system with itself as leader-follower systems.

Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition

In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order system with line equilibrium. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, through the Lyapunov exponent spectrum and bifurcation graph of the system under the change of the initial value, the influence of fractional order q on the system state can be observed. That is, integer-order systems do not have the phenomenon of attractors coexistence, while fractional-order systems have it.

Fractional-Order Hidden Attractor Based on the Extended Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

Dynamic Analysis, Circuit Design, and Synchronization of a Novel 6D Memristive Four-Wing Hyperchaotic System with Multiple Coexisting Attractors

In this work, a novel 6D four-wing hyperchaotic system with a line equilibrium based on a flux-controlled memristor model is proposed. The novel system is inspired from an existing 5D four-wing hyperchaotic system introduced by Zarei (2015). Fundamental properties of the novel system are discussed, and its complex behaviors are characterized using phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. When a suitable set of parameters are chosen, the system exhibits a rich repertoire of dynamic behaviors including double-period bifurcation of the quasiperiod, a single two-wing, and four-wing chaotic attractors. Further analysis of the novel system shows that the multiple coexisting attractors can be observed with different system parameter values and initial values. Moreover, the feasibility of the proposed mathematical model is also presented by using Multisim simulations based on an electronic analog of the model. Finally, the active control method is used to design the appropriate controller to realize the synchronization between the proposed 6D memristive hyperchaotic system and the 6D hyperchaotic Yang system with different structures. The Routh–Hurwitz criterion is used to prove the rationality of the controller, and the feasibility and effectiveness of the proposed synchronization method are proved by numerical simulations.

Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer

In this work a novel chaotic system with a line equilibrium is presented. First, a dynamical analysis on the system is performed, by computing its bifurcation diagram, continuation diagram, phase portraits and Lyapunov exponents. Then, the system is applied to the problem of secure communication. We assume that the transmitted signal is an additional state. For this reason, the nonlinear system is rewritten in a rectangular descriptor form and then an observer is constructed for achieving synchronization and input reconstruction. If we assume some rank conditions (on the nonlinearities and the solvability of a linear matrix inequality (LMI)) on the system matrices then the observer synchronization can be feasible. We evaluate and demonstrate our approach with specific numerical results.

A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and Circuit Design

<p>A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.</p>