Lobe dynamics and the escape from a potential well

Abstract
Periodically-forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled. The trellises formed by such overlapping manifolds separate the Poincaré plane into regions (calledlobes) whose fates at successive time periods are amenable to analysis. The lobe configurations observed in simple escape systems are discussed, together with their interrelation with the location of periodic orbits and other invariant sets. Three applications of this procedure are illustrated: (i) a numerical technique for localizing the non-wandering set; (ii) a demonstration that the existence of a saddle possessing aboundedhomoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction; (iii) the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle.

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