Existence and Stability of Periodic Orbits in $N$-Dimensional Piecewise Linear Continuous Maps

TL;DR

This paper introduces an efficient matrix-based method to determine the existence and stability of periodic orbits in high-dimensional piecewise linear maps, avoiding brute-force iterative procedures.

Contribution

The authors develop a novel approach for analyzing periodic orbits in piecewise linear maps by directly computing matrix powers, applicable to any dimension, and demonstrate its efficiency in three-dimensional systems.

Findings

The proposed method outperforms traditional algorithms in higher dimensions.

It enables faster identification of existence and stability regions of periodic orbits.

Application to 3D maps illustrates practical utility and effectiveness.

Abstract

Piecewise smooth maps are known to exhibit a wide range of dynamical features including numerous types of periodic orbits. Predicting regions in parameter space where such periodic orbits might occur and determining their stability is crucial to characterize the dynamics of the system. However, obtaining the conditions of existence and stability of these periodic orbits generally use brute force methods which require successive application of the iterative map on a starting point. In this article, we propose a faster and more elegant way of obtaining those conditions without iterating the complete map. The method revolves around direct computation of higher powers of matrices without computing the lower ones and is applicable on any dimension of the phase space. In the later part of the article, we compare the speed of the proposed method with the other popular algorithms which shows the effectiveness of the proposed method in higher dimensions. We also illustrate the use of this method in computing the regions of existence and stability of a particular class of periodic orbits in three dimensions.