Stable periodic motions in the problem on passage through a separatrix

Abstract

A Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.(c) 1997 American Institute of Physics.

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

“…Statistical independence follows from the divergence of phases along trajectories. For separatrix crossings consecutive crossings for some initial conditions are statistically dependent (Cary and Skodje, 1989) and islands of stability, albeit being of a small measure, do exist (Elskens and Escande, 1991;Neishtadt et al, 1997) inside large chaotic see. Consecutive resonance crossings should be treated as independent as shows the following reasoning from (Neishtadt, 1999).…”

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

“…The existence of stability islands with total measure not small with ε in the domain of separatrix crossings was established in [10,11] in the case of a Hamiltonian system with one degree of freedom and the Hamiltonian function slowly periodically depending on time. Here we generalize this result to 2 d.o.f.…”

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

“…Our numerical simulations, which show the same scaling that our theoretical results indicate, suggest the presence of (parameter-dependent) integrable dynamics of positive but small measure inside the "chaotic sea." [48] Remark 11 : A similar combination of islands of invariant tori within a chaotic sea occurs in the example of a parametrically forced planar pendulum [7]. The division of phase space into a mostly quasiperiodic and a mostly chaotic region is the typical behavior that one expects to observe in a large class of forced one dof Hamiltonian systems [15].…”

“…In the near-autonomous setting (i.e., for small-amplitude V ), the figures show large domains of integrable dynamics in the interior region. A perturbation analysis for a similar system shows that, as the amplitude of V goes to infinity, the sizes of these islands vanish, but their number increases reciprocally, so that an integrable set of measure O(1) remains [48].…”