Some remarks on the abundance of stable periodic orbits inside homoclinic lobes

PHYSICA D-NONLINEAR PHENOMENA

卷:240

Article

Simo, C.¹  Vieiro, A.¹ 

[1] Univ Barcelona, Dept Matemat Aplicada & Anal, E-08007 Barcelona, Spain

关键词: Dynamical systems;    Return maps;    Stability islands;    Chaos;    Ergodicity;   

DOI  :  10.1016/j.physd.2011.09.007

来源: Elsevier

【 摘 要 】

We consider a family F-epsilon of area-preserving maps (APMs) with a hyperbolic point H-epsilon whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map (SM) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families F-epsilon with central symmetry and close to an integrable limit, of the relative measure of the set of parameters e for which F-epsilon has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families F-epsilon of APMs: The SM family, and we compare our analytical results with the numerical estimates. The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form. The conservative Henon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands. The results obtained can be seen as the quantitative analogs to those in Simo and Treschev (2008) [9], although here we deal with the a priori stable situation instead. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free