Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods

In this work, a powerful analytical method, called He’s parameter-expanding methods (HPEM) is used to obtain the exact solutions of non-linear modified Van der Pol’s oscillator. The classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Comparison of the obtained solution with those obtained using perturbation method shows that this method is effective and convenient to solve this problem. This method introduces a capable tool to solve this kind of non-linear problems.     

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

This paper aims to investigate the stochastic response of the van der Pol(VDP) oscillator with two kinds of fractional derivatives under Gaussian white noise excitation.First,the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique.Then,the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution.Finally,the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator.The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order,the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator.An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.     

Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions

This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincaré section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.     

Nonlinear parametric oscillations in certain stochastic systems: A random van der Pol oscillator

Astochastic model for some class ofnonlinear oscillators, which includes a van der Pol-type oscillator with random parameters, is analyzed in thediffusion limit. That is, small random fluctuations and long time are considered, while the nonlinearity is also assumed to be small. We show that there existstationary distributions, independent of the phase of the oscillator, a result proved earlier by R. L. Stratonovich assuming the random perturbations of the frequency to be delta correlated. The time behavior of the moments of the displacement of the oscillator from its rest position is also investigated and the results are compared with the corresponding ones for the linear random oscillator. A numerical study is also performed for the first two moments and plots are given.on leave from the University of Padua, Italy.