Deterministic Chaos: Sensitivity, Mixing, and Periodic Points

Abstract

Mathematical research in chaos can be traced back at least to 1890, when Henri Poincaré studied the stability of the solar system. He asked if the planets would continue on indefinitely in roughly their present orbits, or might one of them wander off into eternal darkness or crash into the sun. He did not find an answer to his question, but he did create a new analytical method, the geometry of dynamics. Today his ideas have grown into the subject called topology, which is the geometry of continuous deformation. Poincaré made the first discovery of chaos in the orbital motion of three bodies which mutually exert gravitational forces on each other.

A dictionary definition of chaos is a ‘disordered state of collection; a confused mixture’. This is an accurate description of dynamical systems theory today — or of any other lively field of research.

Morris W. Hirsch¹