M. Siewe

Nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ6-rayleigh oscillator combined to parametric excitations

The nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ 6 -Rayleigh oscillator combined to parametric excitations is studied in this paper. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. The original oscillator is transformed to averaged equations using the method of harmonic balance to obtain periodic solutions. The response curves show the saddle-node bifurcation and the multi-stability phenomena. Based on the Melnikovs method, horseshoe chaos is found and its control is made by introducing an external periodic perturbation.

Investigating bifurcations and chaos in magnetopiezoelastic vibrating energy harvesters using Melnikov theory

In this paper, we present a bifurcation and chaos analysis of a magnetopiezoelastic vibration energy harvester including inductance. We explain the behavior of these energy harvesters, particularly in the chaotic regime. We analyzed the model by using the Melnikov method, bifurcation diagrams, Lyapunov exponents and spectral methods. We determined the conditions for the onset of transition to chaos. This allows us to bound the regions of control parameters where the system displays desired chaotic oscillations and thus characterize the maximal harvestable power for this particular architecture.

TRANSITION TO CHAOS IN PLASMA DENSITY WITH ASYMMETRY DOUBLE-WELL POTENTIAL FOR PARAMETRIC AND EXTERNAL HARMONIC OSCILLATIONS

We study the chaotic dynamics of one-degree-of-freedom nonlinear oscillator representing a density perturbation in plasma device model excited by parametric and external driven forces. Critical parameters for the onset of chaotic motions are specified using Melnikov method. The analytical results are confirmed by numerical simulations. The global dynamical changes of the system have been examined by evaluating parametric changes of the bifurcation diagrams, maximum Lyapunov exponent, Poincare map and the basin boundaries of attraction. The transitions to chaos caused by the cascade bifurcation and intermittency are clearly shown by graphical methods.

Nonlinear dynamics of parametrically driven particles in a Φ6 potential

A general parametrically excited mechanical system is considered. Approximate solutions are determined by applying the method of multiple time scales. It is shown that only combination parametric resonance of the additive type is possible for the system examined. For this case, the existence and stability properties of the fixed points of the averaged equations corresponding to the nontrivial periodic solutions of the original system are investigated. Thus, emphasis is placed on understanding the chaotic behaviour of the extended Duffing oscillator in the Φ6 potential under parametric excitation for a specific parameter choice. From the Melnikov-type technique, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation. Our analysis is carried out in the case of a triple well with a double hump which does not lead to unbounded motion; this analysis is complemented by numerical simulations from which we illustrate the fractality of the basins of attraction. The results show that the threshold amplitude of parametric excitation moves upwards as the parametric intensity increases. Numerical simulations including bifurcation diagrams, Lyapunov exponents, phase portraits and Poincare maps are shown.