Existence of Unpredictable Solutions and Chaos

TL;DR

This paper introduces a novel approach to chaos by defining unpredictable solutions as single functions in differential equations, demonstrating their existence and extending the concept to various systems including Duffing equations.

Contribution

It is the first work to describe chaos originating from a single unpredictable function rather than a collection of functions.

Findings

Existence of unpredictable solutions in quasi-linear differential equations.

Chaotic behavior can be initiated from a single unpredictable function.

Application demonstrated on Duffing equations.

Abstract

In paper [1] unpredictable points were introduced based on Poisson stability, and this gives rise to the existence of chaos in the quasi-minimal set. This time, an unpredictable function is determined as an unpredictable point in the Bebutov dynamical system. The existence of an unpredictable solution and consequently chaos of a quasi-linear system of ordinary differential equations are verified. This is the first time that the description of chaos is initiated from a single function, but not on a collection of them. The results can be easily extended to different types of differential equations. An application of the main theorem for Duffing equations is provided.