J. Kengne

Coexistence of Multiple Attractors and Crisis Route to Chaos in a Novel Chaotic Jerk Circuit

In this paper, a novel autonomous RC chaotic jerk circuit is introduced and the corresponding dynamics is systematically investigated. The circuit consists of opamps, resistors, capacitors and a pair of semiconductor diodes connected in anti-parallel to synthesize the nonlinear component necessary for chaotic oscillations. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a linear transformation of model MO15 previously introduced in [Sprott, 2010]. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. The bifurcation analysis indicates that chaos arises via the usual paths of period-doubling and symmetry restoring crisis. One of the key contributions of this work is the finding of a region in the parameter space in which the proposed (“elegant”) jerk circuit exhibits the unusual and striking feature of multiple attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). Laboratory experimental results are in good agreement with the theoretical predictions.

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincare sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the pro...

Theoretical Analysis and Adaptive Synchronization of a 4D Hyperchaotic Oscillator

We propose a new mathematical model of the TNC oscillator and study its impact on the dynamical properties of the oscillator subjected to an exponential nonlinearity. We establish the existence of hyperchaotic behavior in the system through theoretical analysis and by exploiting electronic circuit experiments. The obtained numerical results are found to be in good agreement with experimental observations. Moreover, the new technique on adaptive control theory is used on our model and it is rigorously proven that the adaptive synchronization can be achieved for hyperchaotic systems with uncertain parameters.

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study

In this chapter, the dynamics of chaotic systems with multiple coexisting attractors is addressed using the well-known Newton–Leipnik system as prototype. In the parameters space, regions of multistability (where the system exhibits up to four disconnected attractors) are depicted by performing forward and backward bifurcation analysis of the model. Basins of attraction of various coexisting attractors are computed, showing complex basin boundaries. Owing to the fractal structure of basin boundaries, jumps between coexisting attractors are predicted in experiment. A suitable electrical circuit (i.e., analog simulator) is designed and used for the investigations. Results of theoretical analysis are verified by laboratory experimental measurements. In particular, the hysteretic behavior of the model is observed in experiment by monitoring a single control resistor. The approach followed in this chapter shows that by combining both numerical and experimental techniques, one can gain deep insight into the dynamics of chaotic systems exhibiting multiple attractor behavior.

Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators

The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos.

A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a ‘hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

Complex Dynamical Behavior of a Two-Stage Colpitts Oscillator with Magnetically Coupled Inductors

A five-dimensional (5D) controlled two-stage Colpitts oscillator is introduced and analyzed. This new electronic oscillator is constructed by considering the well-known two-stage Colpitts oscillator with two further elements (coupled inductors and variable resistor). In contrast to current approaches based on piecewise linear (PWL) model, we propose a smooth mathematical model (with exponential nonlinearity) to investigate the dynamics of the oscillator. Several issues, such as the basic dynamical behaviour, bifurcation diagrams, Lyapunov exponents, and frequency spectra of the oscillator, are investigated theoretically and numerically by varying a single control resistor. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling and interior crisis routes as the single control resistor is monitored. Furthermore, an experimental study of controlled Colpitts oscillator is carried out. An appropriate electronic circuit is proposed for the investigations of the complex dynamics behaviour of the system. A very good qualitative agreement is obtained between the theoretical/numerical and experimental results.