C. Tchawoua

Stochastic Bifurcations induced by correlated Noise in a Birhythmic van der Pol System

Abstract We investigate the effects of exponentially correlated noise on birhythmic van der Pol type oscillators. The analytical results are obtained applying the quasi-harmonic assumption to the Langevin equation to derive an approximated Fokker–Planck equation. This approach allows to analytically derive the probability distributions as well as the activation energies associated to switching between coexisting attractors. The stationary probability density function of the van der Pol oscillator reveals the influence of the correlation time on the dynamics. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that noise intensity and correlation time can be treated as bifurcation parameters. Comparing the analytical and numerical results, we find good agreement both when the frequencies of the attractors are about equal or when they are markedly different.

Chirped self-healing Airy pulses compression in silicon waveguides under fourth-order dispersion

Abstract We present the compression of Airy pulses in silicon-on-insulator (SOI) waveguides under the fourth-order dispersion (FOD) using the variational approach that involves Rayleigh’s dissipation function (RDF). All the pulse characteristics are under the influence of the two-photon and the frequency-carrier absorptions. In a quasi-linear approximation, the pulse compression conditions induced by the interaction of the group-velocity dispersion (GVD), the chirp and the FOD are derived. In the nonlinear case, the self-phase modulation (SPM), the two-photon absorption (TPA) and the free-carrier absorption (FCA) reduce the length of compression in a propagation regime of normal GVD, positive chirp and a negative value of FOD. The TPA reduces the maximal power reached than the SPM while the FCA rather increases its value. These results are confirmed in the general case where they all interact with the linear dispersion terms of the SOI waveguide.

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.

Pseudopotential of birhythmic van der Pol-type systems with correlated noise

We propose to compute the effective activation energy, usually referred to a pseudopotential or quasipotential, of a birhythmic system—a van der Pol-like oscillator—in the presence of correlated noise. It is demonstrated, with analytical techniques and numerical simulations, that the correlated noise can be taken into account and one can retrieve the low noise rate of the escapes. We thus conclude that a pseudopotential, or an effective activation energy, is a realistic description for the stability of birhythmic attractors also in the presence of correlated noise.

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.

Evaluation of Vapor Pressure Estimation Methods for Use in Simulating the Dynamic of Atmospheric Organic Aerosols

The modified Mackay (mM), the Grain-Watson (GW), Myrdal and Yalkovsky (MY), Lee and Kesler (LK), and Ambrose-Walton (AW) methods for estimating vapor pressures () are tested against experimental data for a set of volatile organic compounds (VOC). required to determine gas-particle partitioning of such organic compounds is used as a parameter for simulating the dynamic of atmospheric aerosols. Here, we use the structure-property relationships of VOC to estimate . The accuracy of each of the aforementioned methods is also assessed for each class of compounds (hydrocarbons, monofunctionalized, difunctionalized, and tri- and more functionalized volatile organic species). It is found that the best method for each VOC depends on its functionality.

HOMOCLINIC BIFURCATION AND CHAOS IN Φ6-RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN BY AN AMPLITUDE MODULATED FORCE

With amplitude modulated excitation, the effect on chaotic behavior of Φ6-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincare map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

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.

Chaos Control and Synchronization of a Class of Uncertain Chaotic Systems

This paper deals with the control and synchronization of chaotic systems. First, a control strategy is developed to control a class of uncertain nonlinear systems. The proposed strategy is an input-output control scheme, which comprises an uncertainty estimator and an exponential linearizing feedback. Computer simulations are provided to illustrate the operation of the designed synchronization scheme.