Description of strange attractors using invariants of phase-plane

摘要

Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory. Attractors are parts of phase space of the dynamical system. There can be many geometrical sets that are attractors. When these sets are hard to describe, then the attractor is a strange attractor. Like snowflakes, these strange attractors come in infinite variety with no two the same. The aim of this paper is to propose a slightly different way to "see" a strange attractor. Three mean quantities are defined and the chaotic motion of the Ueda oscillator, the simplest quadratic oscillator and the Rucklidge oscillator is analyzed in order to draw the so-called invariants of phase-plane, who may be regarded as "marks" of the strange attractors.