Stability islands in the vicinity of separatrices of near-integrable symplectic maps

Carles Simó; Dmitry Treschev

doi:10.3934/dcdsb.2008.10.681

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2008, Volume 10, Issue 2&3: 681-698. Doi: 10.3934/dcdsb.2008.10.681

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Stability islands in the vicinity of separatrices of near-integrable symplectic maps

Carles Simó^1, and
Dmitry Treschev^2,

1.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona
2.
Steklov Mathematical Institute, Gubkina str., Moscow, 119991

Received: November 30, 2006

Revised: April 30, 2007

Published: August 2008

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Abstract

We discuss the problem of existence of elliptic periodic trajectories inside lobes bounded by segments of stable and unstable separatrices of a hyperbolic fixed point. We show that such trajectories generically exist in symplectic maps arbitrary close to integrable ones. Elliptic periodic trajectories as a rule, generate stability islands. The area of such an island is of the same order as the lobe area, but the quotient of areas can be very small. Numerical examples are included.

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Mathematics Subject Classification: Primary: 37J40; Secondary: 70K44, 70H08.

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