Finding, Stabilizing, and Verifying Cycles of Nonlinear Dynamical Systems

TL;DR

This paper introduces a novel control method for nonlinear dynamical systems that effectively finds, verifies, and stabilizes unknown cycles by leveraging geometric complex functions, extending previous one-dimensional solutions to multi-dimensional spaces.

Contribution

The paper presents a new control approach based on mixing past states, enabling stabilization and discovery of cycles in multi-dimensional systems, improving upon existing linear algebra methods.

Findings

Successfully stabilizes unknown cycles in nonlinear systems

Extends one-dimensional solutions to multi-dimensional spaces

Provides practical algorithms with numerical examples

Abstract

We present a new solution for fundamental problems in nonlinear dynamical systems: finding, verifying, and stabilizing cycles. The solution we propose consists of a new control method based on mixing previous states of the system (or the functions of these states). This approach allows us to locally stabilize and to find a priori unknown cycles of a given length. Our method generalizes and improves on the existing one dimensional space solutions to multi-dimensional space while using the geometric complex functions theory rather than a linear algebra approach. Several numerical examples are considered. All statements and formulas are given in final form. The formulas derivation and reasoning may be found in the cited references. The article focuses on practical applications of methods and algorithms.