Bistable oscillating states in dissipative dynamical systems: Scaling properties and one-dimensional maps

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Abstract:

A variety of dynamical systems displaced far from equilibrium exhibits bistability and hysteresis involving oscillating states. For strong dissipation the flows for such systems admit a reduced description in terms of discrete-time, one-dimensional maps: the infinitely dissipative limit. Classes of such maps producing isolated cusps or infinite sets of cusps are compared. Where infinite sets of such bistabilties exist they obey universal scaling laws. The persistence of such structures at finite dissipation, i.e., in two-dimensional maps which can exactly represent the flow, is also examined. Some of the implications of the results for experimental studies on these dynamical systems are discussed. © 1984 American Chemical Society.