Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor

Details Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor

2009-02-04

arxiv.org/abs/0902.0681v1

Mathematics

Dynamical Systems

41 pages, no figures

Scientific paper

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.

Affiliated with

Also associated with

No associations

Rating

Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-553319