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A universal circuit for studying and generating chaos. I: Routes to chaos. (English) Zbl 0844.58052
Summary: In this introductory tutorial paper, we demonstrate the generality of Chua’s oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua’s oscillator can generate the same qualitative behavior as any member of a 21-parameter family 𝒞 of continuous, odd-symmetric, piecewise linear vector field in 3 . This result is of fundamental importance because it unifies many previously published papers on chaotic circuits and systems (e.g. examples from Brockett, Sparrow, Arnéodo, Nishio, Ogorzalek, etc.) under one umbrella, thereby obviating the need to analyze these circuits and systems as separate and unrelated systems. Indeed, every bifurcation and chaotic phenomena exhibited by any member of the family 𝒞 is also exhibited by this universal circuit. In a companion paper [part II: the authors, ibid., 745-761 (1993; see the paper below)], we show how the generality of Chua’s oscillator can be used to approximate other chaotic systems in the literature which are not necessarily piecewise linear.

MSC:
37D45Strange attractors, chaotic dynamics
34C28Complex behavior, chaotic systems (ODE)