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An adaptive fuzzy system for modeling chaos. (English) Zbl 0827.58039
Summary: Most chaotic systems do not have even dynamical behavior over the whole phase space. In some regions, the system stretches and branches more violently than others. In the regions where the dynamics are violent, finer representation must be given. In this paper, we make use of this property to model chaos. We present an adaptive system based on fuzzy logic. It can refine its representation of a region in the phase space if that region requires it. It does so by adaptively generating more fuzzy rules to model a region only if that region has very violent dynamics. Experiments were performed to test the adaptive fuzzy system for capturing the dynamics of a normal dynamical system (the Van der Pol oscillator) as well as two chaotic systems (the Lorenz and Rossler attractors). Results indicate that the fuzzy system can produce an accurate model of the three dynamical systems. The adaptive rule generation algorithm allowed the fuzzy system to have an optimal number of rules.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
03E72 Theory of fuzzy sets, etc.
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