Asymmetry-Induced Dynamics for a Class of Diode-Based Chaotic Circuits: A Case Study

We consider the modeling and asymmetry-induced dynamics for a class of chaotic circuits sharing the same feature of an antiparallel diodes pair as the nonlinear component. The simple autonomous jerk circuit of [J. Kengne, Z. T. Njitacke, A. N. Nguomkam, M. T. Fouodji and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcation Chaos Appl. Sci. Eng.26 (2016) 1650081] is used as the prototype. In contrast to current approaches where the diodes are assumed to be identical (and thus a perfect symmetric circuit), we examine the more realistic situation where the diodes have different electrical properties in spite of unavoidable scattering of parameters. In this case, the nonlinear component formed by the diodes pair displays an asymmetric current–voltage characteristic which induces asymmetry of the whole circuit. The model is described by a continuous-time 3D autonomous system (ODEs) with exponential nonlinearities. We examine the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectory plots as the main indicators. Period doubling route to chaos, merging crisis, and multiple coexisting (i.e., two, four, or six) mutually symmetric attractors are reported in the symmetric mode of oscillation. In the asymmetric mode, several unusual nonlinear behaviors arise such as coexisting bifurcations, hysteresis, asymmetric double-band chaotic attractor, crisis, and coexisting multiple (i.e., two, three, four, or five) asymmetric attractors for some suitable ranges of parameters. Theoretical analyses and circuit experiments show a very good agreement. The results obtained in this work let us conjecture that chaotic circuits with antiparallel diodes pair are capable of much more complex dynamics than what is reported in the current literature and thus should be reconsidered accordingly in spite of the approach followed in this work.

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.

Dynamical Behavior of a 3D Jerk System with a Generalized Memristive Device

As a new type of electronic components, a memristive device is receiving worldwide attention and can enrich the dynamical behaviors of the oscillating system. In this paper, we propose a 3D jerk system by introducing a generalized memristive device. It is found that the dynamical behaviors of the system are sensitive to the initial conditions even the system parameters are fixed, which results in the coexistence of multiple attractors. And there exists different transition behaviors depending on the selection of the parameters and initial values. Thereby, it is one important type of the candidate system for secure communication since the reconstruction of accurate state space becomes more difficult. Moreover, we build a hardware circuit and the experimental results effectively confirm the theoretical analyses.

Coexistence of Multiple Attractors in an Active Diode Pair Based Chua’s Circuit

This paper focuses on the coexistence of multiple attractors in an active diode pair based Chua’s circuit with smooth nonlinearity. With dimensionless equations, dynamical properties, including boundness of system orbits and stability distributions of two nonzero equilibrium points, are investigated, and complex coexisting behaviors of multiple kinds of disconnected attractors of stable point attractors, limit cycles and chaotic attractors are numerically revealed. The results show that unlike the classical Chua’s circuit, the proposed circuit has two stable nonzero node-foci for the specified circuit parameters, thereby resulting in the emergence of multistability phenomenon. Based on two general impedance converters, the active diode pair based Chua’s circuit with an adjustable inductor and an adjustable capacitor is made in hardware, from which coexisting multiple attractors are conveniently captured.

Circuit Implementation, Synchronization of Multistability, and Image Encryption of a Four-Wing Memristive Chaotic System

The four-wing memristive chaotic system used in synchronization is applied to secure communication which can increase the difficulty of deciphering effectively and enhance the security of information. In this paper, a novel four-wing memristive chaotic system with an active cubic flux-controlled memristor is proposed based on a Lorenz-like circuit. Dynamical behaviors of the memristive system are illustrated in terms of Lyapunov exponents, bifurcation diagrams, coexistence Poincaré maps, coexistence phase diagrams, and attraction basins. Besides, the modular equivalent circuit of four-wing memristive system is designed and the corresponding results are observed to verify its accuracy and rationality. A nonlinear synchronization controller with exponential function is devised to realize synchronization of the coexistence of multiple attractors, and the synchronization control scheme is applied to image encryption to improve secret key space. More interestingly, considering different influence of multistability on encryption, the appropriate key is achieved to enhance the antideciphering ability.

A Time-Delayed Hyperchaotic System Composed of Multiscroll Attractors With Multiple Positive Lyapunov Exponents

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.