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2005, 2005(Special): 846-853. doi: 10.3934/proc.2005.2005.846

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

Bourama Toni 1,

1. 

Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA 23806, United States

Received  September 2004 Revised  March 2005 Published  September 2005

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.
Citation: Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
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