Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

Abstract

In this paper, some aspects on the periodic solutions of the extended Duffing–Van der Pol oscillator are discussed. Doing different rescaling of the variables and parameters of the system associated with the extended Duffing–Van der Pol oscillator, we show that it can bifurcate one or three periodic solutions from a two-dimensional manifold filled by periodic solutions of the referred system. For each rescaling we exhibit concrete values for which these bounds are reached. Beyond that we characterize the stability of some periodic solutions. Our approach is analytical and the results are obtained using the averaging theory and some algebraic techniques.

Keywords:

• extended Duffing–Van der Pol oscillator
• periodic solution
• non-autonomous systems
• averaging theory

2010 AMS Subject Classifications:

• Primary: 34C07
• 34C15
• 34C25
• 34C29
• 37C60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by the FAPESP-BRAZIL grants 2010/18015-6, 2012/05635-1 and 2013/25828-1. The second author is partially supported by MINECO/FEDER grants MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014SGR568, an ICREA Academia, grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378 and a CAPES grant 88881. 030454/2013-01 do Programa CSF-PVE.