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

Author: A.Neishtadt (1), C. Sim\'o (2), D. Treschev (3) and A. Vasiliev (1)

(1) Space Research Institute, Profsoyuznaya 84/32, Moscow 117997, Russia.
(2) Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona,
    Gran Via, 585, Barcelona 08007, Spain.
(3) V.A. Steklov Mathematical Institute, Gubkina 8, Moscow 119991, Russia.
E-mail: aneishta@iki.rssi.ru, carles@maia.ub.es, valex@iki.rssi.ru,
        treschev@mi.ras.ru

Abstract

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom
corresponding to fast motion and the other 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 eps << 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/eps,
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 eps. Thus, the total measure of the
stability islands is estimated from below by a value independent of eps. 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.