Similarity Between Two Dynamical Systems

TL;DR

This paper investigates the degree of similarity between two dynamical systems by analyzing the existence of a homeomorphic map that minimizes a measure of similarity, extending classical theorems and applying to well-known systems.

Contribution

It introduces new conditions for the existence of a similarity measure between dynamical systems and extends the conjugacy concept using Takens embedding theorem.

Findings

Established necessary and sufficient conditions for the similarity minimizer.

Proved a similarity theorem based on Takens embedding theorem.

Validated the approach with simulations on Lorenz, Chua's circuit, and Chen's systems.

Abstract

The main focus of this paper is to explore how much similarity between two dynamical systems. Analogous to the classical Hartman-Grobman theorem, the relationship between two systems can be linked by a homeomorphic map $K$, and the core is to study the minimizer $K^*$ to measure the degree of similarity. We prove the sufficient conditions and necessary conditions (the maximum principle) for the existence of the minimizer $K^*$. Further, we establish similarity theorem based on the Takens embedding theorem. As applications, Lorenz system, Chua's circuit system and Chen's system are simulated and tested. The results illustrate what is the similarity, which extends the conjugacy in dynamical systems.