Formulas for the amplitude of the van der Pol limit cycle

Details Formulas for the amplitude of the van der Pol limit cycle Formulas for the amplitude of the van der Pol limit cycle

2008-06-10

arxiv.org/abs/0806.1634v1

Nonlinear Sciences

Adaptation and Self-Organizing Systems

13 pages, 5 figures, 1 table

Scientific paper

The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $\epsilon$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.

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