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September  2008, 10(2&3, September): 681-698. doi: 10.3934/dcdsb.2008.10.681

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  December 2006 Revised  May 2007 Published  June 2008

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.
Keywords: lobes defined by invariant manifold, area preserving maps, separatrix maps., stability islands.
Mathematics Subject Classification: Primary: 37J40; Secondary: 70K44, 70H0.
Citation: Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681
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