Stability and dynamics of nonautonomous systems with pulsed nonlinearity

Abstract

We study the dynamics of a class of nonautonomous systems with pulsed nonlinearity that consist of a periodic sequence of linear and nonlinear autonomous systems, each one acting alone in a different time or space interval. We focus on the investigation of control capabilities of such systems in terms of altering their fundamental dynamical properties by appropriate parameter selections. For the case of single oscillators, the stability of the zero solution as well as the phase space topology is shown to drastically depend on parameters such as the frequency of the linear oscillations and the durations of the linear and nonlinear intervals. In cases of chain of coupled oscillators with pulsed onsite nonlinearity, it is shown that appropriate parameter selections can stabilize an otherwise unstable zero background allowing for the existence of dynamically robust localized excitations, whose evolution properties can now be explicitly determined and controlled.