The chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations are investigated both analytically and numerically in this paper. The critical curves separating the chaotic and nonchaotic regions are obtained. The chaotic feature on the system parameters is discussed in detail. Some new dynamical phenomena including the controllable frequency are presented for this system. Numerical results are given, which verify the analytical ones.

Dufffing-Van der Pol oscillator has a wide usage in many fields. The dynamics of Duffing-Van der Pol oscillator has been investigated widely in these years. Using the Melnikov method, Ravisankar et al. [

In this paper, the chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations are studied analytically with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are obtained. The chaotic feature on the system parameters is discussed in detail and some new dynamical phenomena are presented. Numerical simulations verify the analytical results.

Consider the Duffing-Van der pol oscillator with external and parametric excitations

Assume the damping and excitation terms

Using the transformations

When

System (

There exist homoclinic orbits connecting

The phase portrait of system (

Melnikov method [

First, taking

The critical curves for chaotic motions of system (

Secondly, taking

The critical curves for chaotic motions of system (

From Figures

For the case of periodic external excitation, the critical curves have the classical bell shape; this means that, with the excitations possessing sufficiently small or very large periods, the systems are not chaotically excited. When

For the case of parametric excitations, the critical curve first decreases quickly to zero and then increases; at last it decreases to zero as

First, choosing the system parameters

The theoretical and numerical critical values for chaos of system (

Next, choosing the system parameters

Using the Melnikov and numerical methods, the chaotic motions for the Duffing-Van der Pol oscillator with external and parametrical excitations are investigated in this paper. The critical curves separating the chaotic and nonchaotic regions are obtained. It is shown that there exists a controllable frequency

The authors declare that there is no conflict of interests regarding the publication of this paper.

The project was supported by National Science Foundation of China (11202095 and 11172125), China Postdoctoral Science Foundation (2013T60531), and the Science Foundation of NUAA (NZ2013213).