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Some remarks on the abundance of stable periodic orbits inside homoclinic lobes. (English) Zbl 1244.37039

Summary: We consider a family \(F_{\varepsilon }\) of area-preserving maps (APMs) with a hyperbolic point \(H_{\varepsilon }\) 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 \(F_{\varepsilon }\) with central symmetry and close to an integrable limit, of the relative measure of the set of parameters \(\varepsilon \) for which \(F_{\varepsilon }\) 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 \(F_{\varepsilon }\) of APMs:
\(\bullet \) The SM family, and we compare our analytical results with the numerical estimates.
\(\bullet \) 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.
\(\bullet \) The conservative Hénon 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 [C. Simó and D. Treschev, Discrete Contin. Dyn. Syst., Ser. B 10, No. 2–3, 681–698 (2008; Zbl 1153.37406)], although here we deal with the a priori stable situation instead.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37D10 Invariant manifold theory for dynamical systems

Citations:

Zbl 1153.37406

Software:

PARI/GP
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References:

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