Bifurcation of critical periods from Pleshkan's isochrones

Details Bifurcation of critical periods from Pleshkan's isochrones Bifurcation of critical periods from Pleshkan's isochrones

2008-11-14

arxiv.org/abs/0811.2317v1

Mathematics

Dynamical Systems

18 pages, 1 figure

Scientific paper

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities $\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\mathscr C_3,$ then at most two critical periods bifurcate from its period annulus. Moreover we show that, for each $k=0,1,2,$ there are perturbations giving rise to exactly $k$ critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers $\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside $\mathscr C_2,$ then at most one critical period bifurcates from its period annulus. In addition, for each $k=0,1,$ we show that there are perturbations giving rise to exactly $k$ critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.

Affiliated with

Also associated with

No associations

Rating

Bifurcation of critical periods from Pleshkan's isochrones 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 Bifurcation of critical periods from Pleshkan's isochrones, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bifurcation of critical periods from Pleshkan's isochrones will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-11058