Analysis of Stable Periodic Orbits in 1-D Linear Piecewise Smooth Maps

TL;DR

This paper provides a complete analytical characterization of stable periodic orbits in 1-D linear piecewise smooth maps, revealing relationships between orbit cardinality and period, and introducing algorithms for pattern enumeration and parameter range determination.

Contribution

It introduces a comprehensive analytic framework for stable periodic orbits and new algorithms for pattern enumeration and parameter range analysis in piecewise smooth maps.

Findings

Established the relationship between orbit cardinality and period.

Proved the existence of (n) distinct admissible patterns for each orbit size n.

Developed algorithms for pattern enumeration and parameter range calculation.

Abstract

By varying a parameter of a one-dimensional piecewise smooth map, stable periodic orbits are observed. In this paper, complete analytic characterization of these stable periodic orbits is obtained. An interesting relationship between the cardinality of orbits and their period is established. It is proved that for any $n$, there exist $\phi(n)$ distinct admissible patterns of cardinality $n$. An algorithm to obtain these distinct admissible patterns is outlined. Additionally, a novel algorithm to find the range of parameter for which the orbit exists is proposed.