Stable periodic motions in the problem on passage through a separatrix

Area of Science:

• * Hamiltonian dynamics
• * Non-linear systems
• * Mathematical physics

Background:

• * Hamiltonian systems with one degree of freedom are analyzed.
• * These systems depend on a slowly, periodically time-varying parameter.
• * Separatrices exist for each fixed parameter value and pulse periodically over time.

Purpose of the Study:

• * To investigate the nature of motion within regions swept by pulsing separatrices in Hamiltonian systems.
• * To determine if these regions exhibit true chaos or structured dynamics.
• * To explore the impact of additional system symmetry on the observed dynamics.

Main Methods:

• * Theoretical analysis of Hamiltonian systems with time-varying parameters.
• * Examination of phase portraits and the behavior of separatrices.
• * Numerical experiments to observe system dynamics in regions swept by separatrices.
• * Analytical investigation of systems with additional symmetry, such as a pendulum in a varying gravitational field.

Main Results:

• * Numeric experiments suggest chaotic motion in regions swept by pulsing separatrices.
• * Systems with additional symmetry (e.g., pendulum) exhibit numerous periodic solutions within these regions.
• * These periodic solutions are surrounded by stability islands.
• * The total measure of these stability islands does not vanish as the parameter variation rate approaches zero.

Conclusions:

• * The presence of symmetry fundamentally alters the dynamics of Hamiltonian systems with time-varying parameters.
• * Regions that appear chaotic due to pulsing separatrices can contain significant stable, periodic motion.
• * Stability islands within these regions are a robust feature, persisting even with slow parameter changes.