We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation. Using perturbation methods we have found a critical forcing amplitude above which the system may behave chaotically. The results have been verified by numerical simulations using standard nonlinear to...

This paper deals with the dynamics of a model describing systems eonsisting of the classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. Both the forced and autonomous cases are considered. Harmonic response is investigated along with its stability boundaries. Condition for quenching phenomena in the autonomous case is derived. Neimark bifurcation is observed and it is...

We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic different physical situations. The system studied with a smaller and larger parameter value, that we call underdamped overdamped regimes. In both have relation between parameter, amplitude oscillations times to reach asymptotic behavior, called times. regime, study shows that, also here, there...

In this paper we have chosen to work with two different approaches solving the inverse problem of calculus variation. The first approach is based on an integral representation Lagrangian function that uses equation motion while second one relies a generalization well known Noether's theorem and constructs directly from motion. As application provide some useful remarks for modified Emden-type t...

We study the characteristics of stochastic resonance (SR) in the Duffing oscillator with three types of asymmetries in its double-well potential. The asymmetries controlled by a parameter α are introduced in the potential by varying (i) the depth of the left-well alone, (ii) the location of the minimum of the left-well alone and (iii) both depth and location of the minimum of the left-well alon...

Chaotic systems are widely used to detect the weak signal of the noise background. The effect of the noise in the detection is often ignored because of the characteristics of sensitive to certain signals and inert to noise. However, it's found that chaotic system has low noise immunity for some different variance of noise after a large number of experiments, which makes mistake for the signal t...

Motivated by recent experimental progress to readout quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing oscillator, are analyzed. In the regime of weak dissipation a consistent diffusion equation in the semiclassical limit is derived to capture the intimate relation be...

In this paper, a methodology for propagation of uncertainty in stochastic nonlinear dynamical systems is investigated. The process noise is approximated using KarhunenLoève (KL) expansion. Perron-Frobenius (PF) operator is used to predict the evolution of uncertainty. A multivariate Kolmogorov-Smirnov test is used to verify the proposed framework. The method is applied to predict uncertainty ev...

The chaotic synchronization of third-order systems and second-order driven oscillator is studied in this paper. Such a problem is related to synchronization of strictly different chaotic systems. We show that dynamical evolution of second-order driven oscillators can be synchronized with the canonical projection of a third-order chaotic system. In this sense, it is said that synchronization is ...