Analysis of stable periodic orbits in the one dimensional linear piecewise-smooth discontinuous map

In this paper, we consider one dimensional linear piecewise-smooth discontinuous maps. It is well known that stable periodic orbits exist for such maps, in some parameter region. It is also known that the corresponding bifurcation phenomena (termed as period adding bifurcation) exhibit a special structure. In the last couple of years, several authors have analyzed this structure using border collision bifurcation curves and given the characterization for various parameter regions. In this paper, we have analyzed a specific parameter range employing a different approach. We show that this approach enables one to pose some interesting questions like: what is the number of distinct periodic orbits of any given cardinality? We prove that there are precisely φ(n) distinct orbits of period n, where φ is the Euler's totient function. We propose an algorithm which calculates the location of fixed points of all these φ(n) distinct orbits and gives the precise range of existence of these orbits with respect to the parameters. Further, we show how the amount of computations required to find these ranges of existence can be optimized.