Time delayed piecewise linear Mathieu equation: an analytical and numerical study

In this study, we consider analytical and numerical exploration of the dynamics of a weakly nonlinear piecewise linear (PWL) Mathieu equation with a time delay. A study of such dynamical systems requires finding the time instants of crossing between the two linear states. Moreover, the time delay further complicates the system dynamics by making it infinite-dimensional. In this study, we employ the method of averaging, incorporating non-analytic PWL basis functions along with corresponding algebraic techniques. This approach enables the application of the method of averaging to the resonantly forced, essentially nonlinear PWL Mathieu equation. We herein show that the amplitude evolution and the bifurcation exhibited by the periodic solutions are very well captured by the method of averaging and is a potent tool in studying this class of dynamical systems.