Unpredictable Points and Chaos

TL;DR

This paper introduces the concept of unpredictable points in dynamical systems, demonstrating their role in generating chaos from individual motions, supported by symbolic dynamics examples.

Contribution

It is the first to define chaos originating from a single motion via unpredictable points, expanding the understanding of chaos in dynamical systems.

Findings

Unpredictable points lead to chaos in quasi-minimal sets.

Chaos can be described from a single motion, not just sets of motions.

Symbolic dynamics exemplifies the theoretical results.

Abstract

It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. The existing definitions of chaos are formulated in sets of motions. This is the first time that description of chaos is initiated from a single motion. The theoretical results are exemplified by means of the symbolic dynamics.