Analysis of atypical orbits in one dimensional linear piecewise-smooth discontinuous map

Abstract

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous map are examined analytically. It is shown that, under certain parameter conditions, the map exhibits atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomena occurs. The paper also illustrate that there exists a specific parameter region in which as parameter is varied, there is a smooth transition from stable to unstable periodic orbits. Moreover, we have derived the expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics that exist at this value of parameter is also found out.Mathematics Subject Classification (2020) 39A23 · 39A28 · 39A33