P. Woafo

Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form

A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.

Active control with delay of vibration and chaos in a double-well Duffing oscillator

Abstract This paper deals with the active control of vibration, snap-through instability and horseshoes chaos in a bistable Duffing oscillator. We determine the range of control parameters which leads to a good control. The effect of time-delay between the detection of vibration and action of the control is particularly investigated.

Analysis of tristable energy harvesting system having fractional order viscoelastic material

A particular attention is devoted to analyze the dynamics of a strongly nonlinear energy harvester having fractional order viscoelastic flexible material. The strong nonlinearity is obtained from the magnetic interaction between the end free of the flexible material and three equally spaced magnets. Periodic responses are computed using the KrylovBogoliubov averaging method, and the effects of fractional order damping on the output electric energy are analyzed. It is obtained that the harvested energy is enhanced for small order of the fractional derivative. Considering the order and strength of the fractional viscoelastic property as control parameter, the complexity of the system response is investigated through the Melnikov criteria for horseshoes chaos, which allows us to derive the mathematical expression of the boundary between intra-well motion and bifurcations appearance domain. We observe that the order and strength of the fractional viscoelastic property can be effectively used to control chaos in the system. The results are confirmed by the smooth and fractal shape of the basin of attraction as the order of derivative decreases. The bifurcation diagrams and the corresponding Lyapunov exponents are plotted to get insight into the nonlinear response of the system.

Linear feedback and parametric controls of vibrations and chaotic escape in a Φ6 potential

Abstract The control of vibration amplitude and chaotic escape of an harmonically excited particle in a single well Φ 6 potential is considered. The linear feedback and parametric control strategies are used. The control efficiency on amplitude is found by analysing the behaviour of the amplitude of the controlled system as compared to that of the uncontrolled system. The conditions for inhibition of the chaotic escape are obtained by means of the Melnikov method.

Shilnikov Chaos and Dynamics of a Self-Sustained Electromechanical Transducer

The dynamics of a self-sustained electromechanical transducer is studied. The stability of the critical points is analyzed using the analytic Routh-Hurwitz criterion. Analytic oscillatory solutions are obtained in both the resonant and non-resonant cases. Chaotic behavior is observed using the Shilnikov theorem and from a direct numerical simulation of the equations of motion.

Resonant oscillations and fractal basin boundaries of a particle in a φ6 potential

This paper considers the dynamics of a periodically forced particle in a φ6 potential. Harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. From the Melnikov theory, we derive the analytical criteria for the occurrence of transverse intersections on the surface of homoclinic and heteroclinic orbits both for the three potential well case and a single potential well case. Our analytical investigations are complemented by the numerical simulations from which we show the fractality of the basins of attraction.

Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks

In this paper, we study the phenomenon of cluster synchronization occurring in a shift-invariant set of coupled Ultra-high frequency current-modulated semiconductor lasers in their chaotic regime. We analytically investigate the nature of these cluster patterns through Floquet theory, and we derive the conditions under which the switching procedure between the various cluster states can be led. Numerical simulations are performed to support the analytic approach.

DYNAMICS OF TWO NONLINEARLY COUPLED OSCILLATORS

Two nonlinearly coupled oscillators submitted to an external periodic force are studied. The method of multiple scales is used to find solutions in the resonant and nonresonant cases. The response curves are discussed. A set of bifurcations diagrams are obtained showing period-doubling and torus-breakdown routes to chaos.

Synchronizing modified van der Pol–Duffing oscillators with offset terms using observer design: application to secure communications

This study addresses the adaptive synchronization of the modified van der Pol?Duffing (MVDPD) oscillator with offset terms. From our investigations of the system dynamics, we obtain that the system presents a chaotic behaviour at weak values of the offset parameters. Routh?Hurwitz criteria are used to study the asymptotic stability of the steady states. An adaptive observer design method is applied to achieve synchronization of two identical MVDPD oscillators with offset. Numerical simulations are given to validate the proposed synchronization approach. Moreover, as an application, the proposed scheme is applied to secure communication. Also, simulation results verify the proposed schemes success in the communication application.

Stability and optimization of chaos synchronization through feedback coupling with delay

Abstract We perform the stability and optimization analysis for the (non)delayed synchronization of Duffing-like oscillators, using a retroactive scheme. Stability boundaries are derived through Floquet theory. Critical values for the feedback synchronization coefficient are found. The influence of the delay and of the onset time of the driving upon stability and synchronization time is also analyzed.