Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization

Pages: 846 - 853, Issue Special, August 2005

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Bourama Toni - Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA 23806, United States (email)

Abstract: We discuss polynomial 1-form small perturbation of an isochronous polynomial 1-form in the Pfaffian form $\omega_{\epsilon}=\omega_0+\epsilon \omega$ where $\omega$ is a n-degree polynomial 1-form, $\epsilon$ a small real parameter, and $\omega_0$ an isochronous 1-form with a known birational linearization $T,$ setting $\omega_0$ as the pullback 1-form $T^*\Cal I_0$ of the exact linear isochrone 1-form $\Cal I_0=dH.$ Using recursively the cohomology decompositions of $\omega$ in the related Petrov module, we construct the Bautin-like ideal of the Poincar\'e-Melnikov functions, and study the zeros of Abelian integrals over the ovals $\tilde H=T^*H=r.$ We then stabilize the sequence of the successive Melnikov functions through a multistep reduction of the system coefficients, and determine in terms of the degrees of $\tilde H$ and $\omega$ the overall upper bounds for limit cycles emerging from the polynomial deformation.

Keywords:  Cyclicity, Isochrones, Perturbations, Cohomology Decomposition.
Mathematics Subject Classification:  58F14, 58F21, 34C15, 34C25

Received: September 2004;      Revised: March 2005;      Published: September 2005.