# American Institute of Mathematical Sciences

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

 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.
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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