Analysis of atypical orbits in one-dimensional piecewise-linear discontinuous maps

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous maps are examined analytically. It is shown that, under certain parameter conditions, maps exhibit atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomenon occurs. The paper also illustrates that there exists a specific parameter region where as the parameter is varied, there is a transition from stable to unstable periodic orbits. Moreover, we have derived an expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics concerning this value of parameter is also given.