Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors.

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.

Extreme Multistability and Antimonotonicity in a Shinriki Oscillator with Two Flux-Controlled Memristors

This paper proposes a novel memristive chaotic circuit which originated from a Shinriki oscillator with two flux-controlled memristors of different polarities. This two-memristor-based Shinriki oscillator (TMSO) having a special plane equilibrium is prone to exhibiting the initial-dependent phenomenon of extreme multistability. To investigate its internal dynamics, a third-order dimensionality reduction model is established by utilizing the constitutive relationship of its memristor’s flux and charge. The uncertain plane equilibrium is transfered into some deterministic model that can accurately predict the dynamical evolution of the system, where interesting phenomena of asymmetric bifurcations, extreme multistability and antimonotonicity are detected and analyzed by evaluating the position and stability of the equilibria in the flux–charge model. The simulation is carried out via Multisim to validate the analysis model, and the comparison of the phase trajectories, before and after dimensionality reduction, shows that this oscillator is good for research and practical use.

Symmetric coexisting attractors and extreme multistability in chaotic system

In this paper, a new four-dimensional (4D) chaotic system with two cubic nonlinear terms is proposed. The most striking feature is that the new system can exhibit completely symmetric coexisting bifurcation behaviors and four symmetric coexisting attractors with the same Lyapunov exponents in all parameter ranges of the system when taking different initial states. Interestingly, these symmetric coexisting attractors can be considered as unusual symmetrical rotational coexisting attractors, which is a very fascinating phenomenon. Furthermore, by using a memristor to replace the coupling resistor of the new system, an interesting 4D memristive hyperchaotic system with one unstable origin is constructed. Of particular surprise is that it can exhibit four groups of extreme multistability phenomenon of infinitely many coexisting attractors of symmetric distribution about the origin. By using phase portraits, Lyapunov exponent spectra, and coexisting bifurcation diagrams, the dynamics of the two systems are fully analyzed and investigated. Finally, the electronic circuit model of the new system is designed for verifying the feasibility of the new chaotic system.

A Novel Memristor Chaotic System with a Hidden Attractor and Multistability and Its Implementation in a Circuit

By introducing an ideal and active flux-controlled memristor and tangent function into an existing chaotic system, an interesting memristor-based self-replication chaotic system is proposed. The most striking feature is that this system has infinite line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. In this paper, bifurcation diagrams and Lyapunov exponential spectrum are used to analyze in detail the influence of various parameter changes on the dynamic behavior of the system; it shows that the newly proposed chaotic system has the phenomenon of alternating chaos and limit cycle. Especially, transition behavior of the transient period with steady chaos can be also found for some initial conditions. Moreover, a hardware circuit is designed by PSpice and fabricated, and its experimental results effectively verify the truth of extreme multistability.