Classification of chaos in 3-D autonomous quadratic systems-I. Basic framework and methods

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Original languageEnglish
Pages (from-to)2459-2479
Journal / PublicationInternational Journal of Bifurcation and Chaos
Issue number9
Publication statusPublished - Sept 2006


This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits. © World Scientific Publishing Company.

Research Area(s)

  • Classification of chaos, Heteroclinic orbit, Homoclinic orbit, Ši'lnikov theorem