Optimal escape from potential wells-patterns of regular and chaotic bifurcation

Abstract

Abstract The patterns of bifurcation governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier are studied by numerical simulation. Both the generic asymmetric single-well cubic potential and the symmetric twin-well potential Duffing oscillator are surveyed by varying three parameters: forcing frequency, forcing amplitude, and damping coefficient. The close relationship between optimal escape and nonlinear resonance within the well is confirmed over a wide range of damping. Subtle but significant differences are observed at higher damping ratios. The possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is cmppletely suppressed above a critical level of the damping ratio (about 0.12 for the asymmetric single-well oscillator). Coincidentally, at almost the same level of damping, the optimal escape condition becomes distinct from the apex in the (ω, F ) plane of the bistable regime; this corresponds to the appearance of chaotic attractors which subsume both resonant and non-resonant motions within one well. At higher damping levels, further changes occur involving conversions from chaotic-saddle to regular-saddle bifurcations. These changes in optimal escape phenomena correspond to codimension three bifurcations at exceptional points in the space of three parameters. These bifurcations are described in terms of homoclinic and heteroclinic structures of invariant manifolds, and changes in accessible boundary orbits. The same sequence of codimension three bifurcations is observed in both the twin-well Duffing oscillator and the asymmetric single-well escape equation. Within the codimension three bifurcation patterns governing escape, one particular codimension two global bifurcation involves a chaotic attractor explosion, or interior crisis, compounded with a blue sky catastrophe or boundary crisis of the exploded attractor. This codimension two bifurcation has structure containing a form of predictive power: knowledge of attractor bifurcations in part of the codimension two pattern permits inference of the attractor and basin bifurcations in the remainder. This predictive power is applicable beyond the context of escape from potential wells. Quantitative correlation of bifurcation patterns between the two equations according to simple scaling laws is tested. The unstable periodic orbits which figure most prominently in the major attractor-basin bifurcations are of periods one and three. Their linking is conveniently interpreted by a three-layer spiral horseshoe structure for the folding action in phase space within a well. The structure of this 3-shoe implies a partial ordering among order three subharmonic saddle-node bifurcations. This helps explain the sequence of codimension three bifurcations near optimal escape. Some bifurcational precedence relations are known to follow from the linking of periodic orbits in a braid on a 3-shoe. Additional bifurcational precedence relations follow from a quantitative property of generic potential wells: the dynamic hilltop saddle has a very large expanding multiplier over one cycle of forcing near fundamental resonance. This quantitative property explains the close coincidence of codimension three bifurcations near the suppression of indeterminate outcomes. An experimentalist's approach to identifying the three-layer template structure from time series data is discussed, including a consistency check involving Poincare indices. The bifurcation patterns emerging at higher damling values create favorable conditions for realizing experimental strategies to recognize optimal escape and locate it in parameter space. Strategies based solely on observations of quasi-steady behavior while remaining always within one well are discussed.