OPTIMAL ESCAPE FROM POTENTIAL WELLS - PATTERNS OF REGULAR AND CHAOTICBIFURCATION

The patterns of bifurcation governing the escape of periodically force d oscillations from a potential well over a smooth potential barrier a re studied by numerical simulation. Both the generic asymmetric single -well cubic potential and the symmetric twin-well potential Duffing os cillator are surveyed by varying three parameters: forcing frequency, forcing amplitude, and damping coefficient. The close relationship bet ween optimal escape and nonlinear resonance within the well is confirm ed over a wide range of damping, Subtle but significant differences ar e observed at: higher damping ratios. The possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is comple tely suppressed above a critical level of the damping ratio (about 0.1 2 for the asymmetric single-well oscillator). Coincidentally, at almos t the same level of damping, the optimal escape condition becomes dist inct from the apex in the to, Fl plane of the bistable regime; this co rresponds to the appearance of chaotic attractors which subsume both r esonant and non-resonant motions within one well. At higher damping le vels, further changes occur involving conversions from chaotic-saddle to regular-saddle bifurcations, These changes in optimal escape phenom ena 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 manifold s, and changes in accessible boundary orbits, The same sequence of cod imension 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 particula r codimension two global bifurcation involves a chaotic attractor expl osion, or interior crisis, compounded with a blue sky catastrophe or b oundary crisis of the exploded attractor, This codimension two bifurca tion 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. T his predictive power is applicable beyond the context of escape from p otential wells, Quantitative correlation of bifurcation patterns betwe en the two equations according to simple scaling laws is tested. The u nstable periodic orbits which figure most prominently in the major att ractor-basin bifurcations are of periods one and three. Their linking is conveniently interpreted by a three-layer spiral horseshoe structur e for the folding action in phase space within a well, The structure o f this 3-shoe implies a partial ordering among order three subharmonic saddle-node bifurcations, This helps explain the sequence of codimens ion three bifurcations near optimal escape. Some bifurcational precede nce relations are known to follow from the Inking of periodic orbits i n a braid on a 3-shoe. Additional bifurcational precedence relations f ollow from a quantitative property of generic potential wells: the dyn amic hilltop saddle has a very large expanding multiplier over one cyc le of forcing near fundamental resonance. This quantitative property e xplains the close coincidence of codimension three bifurcations near t he suppression of indeterminate outcomes. An experimentalist's approac h to identifying the three-layer template structure from time series d ata is discussed, including a consistency check involving Poincare ind ices. The bifurcation patterns emerging at higher damping values creat e favorable conditions for realizing experimental strategies to recogn ize 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.