Qualitative bifurcation diagrams

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Abstract

We explore the use of qualitative reasoning to predict the behaviour of a dynamical system given a change in one of its parameters based upon a qualitative representation of its bifurcation diagram. We present three algorithms to perform this task. The first algorithm generates a qualitative representation from a quantitative representation of the bifurcation diagram of the system. The second uses the qualitative representation to simulate the behaviour of the system given a sequence of parameter adjustments or perturbations. The third algorithm solves the opposite, control problem: it determines what parameters to change to take the system from an initial to a given goal state. The first algorithm segments a quantitative bifurcation diagram into MSs having the same qualitative behaviour. These MSs are then interconnected into a relational network. The network is used by the other two algorithms to simulate and plan the behaviour of the system from an initial situation. We present examples illustrating the qualitative representations and the behaviours of the simulation and planning algorithms for dynamical systems of one parameter.

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