Electron. J. Diff. Equ., Vol. 2014 (2014), No. 22, pp. 1-9.

Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation

Xenakis Ioakim

Abstract:
In this article, we study the bifurcation of limit cycles from the linear oscillator $\dot{x}=y$, $\dot{y}=-x$ in the class
$$ \dot{x}=y,\quad \dot{y}=-x+\varepsilon y^{p+1}\big(1-x^{2q}\big), $$
where $\varepsilon$ is a small positive parameter tending to 0, $p \in \mathbb{N}_0$ is even and $q \in \mathbb {N}$. We prove that the above differential system, in the global plane where $p \in \mathbb{N}_0$ is even and $q \in \mathbb{N}$, has a unique limit cycle. More specifically, the existence of a limit cycle, which is the main result in this work, is obtained by using the Poincare's method, and the uniqueness can be derived from the work of Sabatini and Villari [6]. We also investigate and some other properties of this unique limit cycle for some special cases of this differential system. Such special cases have been studied by Minorsky [3] and Moremedi et al [4].

Submitted June 22, 2013. Published January 10, 2014.
Math Subject Classifications: 34C07, 34C23, 34C25.
Key Words: Generalized Van der Pol equation; limit cycles; existence; uniqueness.

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Xenakis Ioakim
Department of Mathematics and Statistics
University of Cyprus
P.O. Box 20537, 1678 Nicosia, Cyprus
email: xioaki01@ucy.ac.cy

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