Stable Manifolds and Separatrices

Gruemm, H.-R. (1975). Stable Manifolds and Separatrices. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-75-139

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Abstract

In the last weeks, many people at IIASA have been concerned with the concept of resilience, after initial work in this direction by C.S. Holling. He talked about resilience as a "measure of the ability of systems to absorb change of state variables, driving variables and parameters and still persist". In my opinion, resilience, thus defined, is directly related to a) the basins of attraction of the system and b) their changes under variations of external parameters. i.e. variations in the time-evolution laws of the system. If one changes the state variables (= the point in phase space describing the system at a given time) but still remains within the same basin, the asymptotic behavior of the system will not change. (This can be made rigorous by a recent theorem of Ruelle and Bowen.) If the dynamics of the system are changed a little, the boundaries of the basins (the separatrices) might move only a little and the structure of the attractors within them might remain the same, such that a point would still trace out a trajectory of the same nature as before the change under the new dynamic laws. On the contrary crossing a separatrix will lead to drastic and catastrophic changes in the long-time behavior of the system, as illustrated in the recent model of Haefele. Haefele proposed therefore that the distance from the next separatrix should be put into a measure of resilience. It is the purpose of this paper to discuss the nature of these separatrices and illustrate their properties in a simple model.

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