Boundaries of Stability: A Potpourri of Dynamic Properties

Jones, D.D. (1973). Boundaries of Stability: A Potpourri of Dynamic Properties. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-73-004

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Abstract

There is more to a system than its equilibrium points. Associated with every stable equilibrium point (or stable limit cycle) is a region of state-space such that any unperturbed trajectory initiated in the region will stay within that region. This is called the region of stability. The boundaries of stability separate contiguous stability regions. An important property of system behavior near these boundaries is that a very small perturbation can move the state of a system across a boundary and transfer the system entirely from one region to another. The system's state cannot move back across the boundary without a subsequent outside perturbation.

The performance of systems near their equilibrium points has been the focus of a considerable amount of investigation. Considerations of optimization, maximization, stable states are examples. The properties of systems far from equilibrium, and particularly near regions of instability (i.e., the boundaries) are not well known.

The significant strategic problem that this paper hopes to address is to locate these boundaries and to determine system dynamics near them. On a tactical level, some approaches are suggested and their usefulness discussed.

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