Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

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September 2008, 10(2&3, September): 621-650. doi: 10.3934/dcdsb.2008.10.621

Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

Anatoly Neishtadt ^1, , Carles Simó ^2, , Dmitry Treschev ^3, and Alexei Vasiliev ^1,

1.	Space Research Institute, Profsoyuznaya 84/32, Moscow 117997, Russian Federation, Russian Federation
2.	Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona
3.	Steklov Mathematical Institute, Gubkina str., Moscow, 119991, Russian Federation

Received November 2006 Revised June 2007 Published June 2008

Abstract
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We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom corresponding to fast motion and the other one corresponding to slow motion. The Hamiltonian function is the sum of potential and kinetic energies, the kinetic energy being a weighted sum of squared momenta. The ratio of time derivatives of slow and fast variables is of order $\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition we prove the existence of many, of order $1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order $\epsilon$. Thus, the total measure of the stability islands is estimated from below by a value independent of $\epsilon$. We find the location of stable periodic trajectories and an asymptotic formula for the number of these trajectories. As an example, we consider the problem of motion of a charged particle in the parabolic model of magnetic field in the Earth magnetotail.

Keywords: slow-fast systems, stability islands., chaotic seas, separatrix crossings.

Mathematics Subject Classification: Primary: 34E10 ; Secondary: 37J4.

Citation: Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621

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