Transition to escape in a system of coupled oscillators
The system consists of two forced pendula, connected by a spring. The spring constant is \(\varepsilon\); the pendula, if uncoupled, are identical. Two notions are introduced: a trajectory is captured if it approaches the resonance manifold and remains in its neighborhood for times of \(O(1/\varepsilon)\); the escaping occurs, when a trajectory passes through the resonance manifold with a time derivative of \(O(1)\), and accordingly its distance from the resonance manifold will be in magnitude \(O(1/\varepsilon)\) after \(O(1/\varepsilon)\) time. There can exist trajectories which, after being captured, escape after times longer than \(O(1/\varepsilon)\) and become unbounded. These motions, called as detained, affect sometimes dramatically the global dynamics. The author applies a perturbation method by utilizing curves of constant potential. Starting with the decoupled \((\varepsilon= 0)\) equations, separatrices are found. Existence and behaviour of these detained motions are analyzed in terms of equipotential surfaces. These provide bounds on the motion of the system, and are shown to be closed for low energies. Above a critical energy level, these curves are open. The detained trajectories are shown to arise in the region of phase space that was (for appropriate energies) stochastic. These motions remain within this region for long times before finally ``leak out'' through an opening in the equipotential curves and go to infinity. Numerous figures visualize the results.
scientific article; zbMATH DE number 1081068