Finding similarity of orbits between two discrete dynamical systems via optimal principle

TL;DR

This paper introduces a novel approach using optimal principles and similarity measures to identify and analyze similarities between orbits of different complex discrete dynamical systems, including chaotic systems.

Contribution

It proposes the concepts of similarity transformation matrix and similarity degree, and applies an optimal principle to quantify orbit similarity in complex dynamical systems.

Findings

Similarity can be detected in chaotic systems like Lorenz and Rössler.

The method reveals rich characteristics and complex behaviors through numerical simulations.

The approach is effective for systems with seemingly irrelevant dynamics.

Abstract

Whether there is similarity between two physical processes in the movement of objects and the complexity of behavior is an essential problem in science. How to seek similarity through the adoption of quantitative and qualitative research techniques still remains an urgent challenge we face. To this end, the concepts of similarity transformation matrix and similarity degree are innovatively introduced to describe similarity of orbits between two complicated discrete dynamical systems that seem to be irrelevant. Furthermore, we present a general optimal principle, giving a strict characterization from the perspective of dynamical systems combined with optimization theory. For well-known examples of chaotic dynamical systems, such as Lorenz attractor, Chua's circuit, R$\rm\ddot{o}$ssler attractor, Chen attractor, L$\rm\ddot{u}$ attractor and hybrid system, with using of the homotopy idea, some numerical simulation results demonstrate that similarity can be found in rich characteristics and complex behaviors of chaotic dynamics via the optimal principle we presented.