Optimized shooting method for finding periodic orbits of nonlinear dynamical systems

TL;DR

This paper introduces a new numerical shooting method leveraging the Levenberg-Marquardt algorithm to efficiently find both stable and unstable periodic orbits in various nonlinear dynamical systems, including autonomous and non-autonomous types.

Contribution

It presents a simple, effective, and easily implementable shooting method that improves accuracy over previous techniques for locating periodic orbits in complex systems.

Findings

Method successfully finds periodic orbits in Rössler and coupled Rössler systems.

Achieves more accurate results compared to previous methods.

Applicable to high-dimensional systems like a flexible rotor-bearing model.

Abstract

An alternative numerical method is developed to find stable and unstable periodic orbits of nonlinear dynamical systems. The method exploits the high-efficiency of the Levenberg-Marquardt algorithm for medium-sized problems and has the additional advantage of being relatively simple to implement. It is also applicable to both autonomous and non-autonomous systems. As an example of its use, it is employed to find periodic orbits in the R\"ossler system, a coupled R\"ossler system, as well as an eight-dimensional model of a flexible rotor-bearing; problems which have been treated previously via two related methods. The results agree with the previous methods and are seen to be more accurate in some cases. A simple implementation of the method, written in the Python programming language, is provided as an Appendix.