H. B. Stewart

BASIN EROSION IN THE TWIN-WELL DUFFING OSCILLATOR: TWO DISTINCT BIFURCATION SCENARIOS

A study is made of the control-phase portraits encountered in the twin-well Duffing oscillator, concentrating on the loss of stability of attractors confined to a single well of the potential. This loss of stability may be understood by examining the bifurcations which precede a single-well attractor touching its basin boundary. Two distinct bifurcation scenarios are described in which the basin boundary develops a fractal structure. This fractal structure accompanies the development of a homoclinic tangency between the inset and outset of the saddle whose inset determines the separatrix action. In the first scenario described, this fractal structure, or chaotic saddle, grows due to a sequence of fractal–fractal basin implosions which are caused by the completion of Smale cycles or heteroclinic chains; the subharmonics involved occur in decreasing order, after the main homoclinic tangency. The second scenario focuses on the bifurcational events which necessarily prepare the creation of a chaotic saddle; the subharmonics involved appear in increasing order, before the main homoclinic event. Both basin erosion scenarios are consistent with a geometric model of the twin-well Duffing oscillator based on a three layer template; in the first scenario, the full three-layer template participates; while in the second scenario, only two layers, a standard horseshoe, play a role.

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

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 experimentalists 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.

Basin explosions and escape phenomena in the twin-well Duffing oscillator: compound global bifurcations organizing behaviour

The sinusoidally drive, twin-well Duffing oscillator has become a central archetypal model for studies of chaos and fractal basin boundaries in the nonlinear dynamics of dissipative ordinary differential equations. It can also be used to illustrate and elucidate universal features of the escape from a potential well, the jumps from one-well to cross-well motions displaying similar characteristics to those recently charted for the cubic one-well potential. We identify here some new codimension-two global bifurcations which serve to organize the bifurcation set and structure the related basin explosions and escape phenomena.

A chaotic blue sky catastrophe in forced relaxation oscillations

Abstract The chaotic attractor of a periodically forced Van der Pol oscillator (Shaw variant) is observed in digital simulation, and is made to vanish in a blue sky catastrophe by increasing a constant (bias) term in the force. The detailed bifurcation diagram, based on extensive simulations, reveals the involvement of the homoclinic outset of a nearby limit cycle of saddle type.

A TUTORIAL GLOSSARY OF GEOMETRICAL DYNAMICS

We present here an informal, encyclopedic glossary of terms and formulae covering the new geometrical concepts of nonlinear dynamics and chaos. It is arranged as an alphabetical dictionary, which can also be read as a connected introduction to the subject by following the go to instructions starting with the entry dynamical system and ending with archetypal equations.

GENERIC PATTERNS OF BIFURCATION GOVERNING ESCAPE FROM POTENTIAL WELLS

Extensive numerical simulations have established the generic character of patterns of bifurcation governing the escape of a forced oscillator across a smooth potential barrier and the close relation with the familiar nonlinear resonance within a potential well. Subtle but significant differences are found, depending on whether the damping is high or low, and the differences are related to the problem of experimentally determining the optimal escape condition.

Folding and mixing in the Birkhoff-Shaw chaotic attractor

Abstract Some new visualization techniques are employed to elucidate the folding topology of the Birkhoff-Shaw chaotic attractor and the mixing action of trajectories wandering across its spirally wing-beak structure.

On the Topological Structure of the Birkhoff-Shaw Strange Attractor

We explore the strange attractor elucidated by Abraham and Shaw governing the chaotic response of a driven Van der Pol equation. The wandering of a trajectory on the attracting bagel is examined.