Analysis of flow hysteresis by a one-dimensional map

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Abstract
The structure of a region of stable period three for a nonlinear dissipative system described by the Rössler equations and a two-parameter cubic map is studied. The intricate configuration of this region, which is bordered by intermittent-type chaos on one side, subharmonic cascades on the other, and possesses several sharp features, is shown to be associated with bistability and hysteresis of the orbits of the flow or map. Locally, the two-parameter cubic map successfully models many features of the differential flow. The mechanism which gives rise to the hysteresis is quite general and corresponds to a cusp catastrophe. This process is described in detail for the map and related to the same phenomenon in the flow.

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