Analysis of Stable Periodic Orbits in the 1-D Linear Piecewise-smooth Discontinuous Map

TL;DR

This paper investigates stable periodic orbits in a 1-D piecewise-smooth discontinuous map, revealing a relationship between orbit period and orbit cardinality, and introduces algorithms for locating and analyzing these orbits efficiently.

Contribution

It introduces a novel method to analyze the relationship between orbit period and cardinality, and proposes algorithms for locating fixed points and their parameter ranges.

Findings

Existence of er(n) orbits for period n

Established a relationship between orbit period and cardinality

Developed algorithms for orbit location and range computation

Abstract

In this paper we consider a one dimensional liner piecewise-smooth discontinuous map. It is well known that stable periodic orbits exist in this type of map for a specific parameter region. It is also known that the corresponding bifurcation phenomena termed as "period adding bifurcation" exhibits a special structure for such case. In last couple of years, various authors have analyzed this structure using border collision bifurcation (BCB) curves and given characterization for various parameter regions. In this paper we have analyzed a specific parameter space using a different method. We show that this method enables us to pose some interesting problems like: is there any relationship between the period of an orbit and the cardinality of orbits? We prove that such a relationship exists and exactly \phi(n) orbits exist for a period n where \phi{} is the Euler's number. We propose an algorithm which gives the location of fixed points of all such \phi(n) distinct orbits and gives their range of existence with respect to parameters. Further, we show that using a novel algorithm the computations to find their range of existence can be reduced.