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Abstract
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.
Mathematics Subject Classification: Primary: 34E10 ; Secondary: 37J40.
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