Calculating the Lyapunov exponents of a piecewise smooth soft impacting system with a time-delayed feedback controller

Abstract

Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise smooth systems with time delayed arguments one faces two difficulties: a high dimension of the discretized state space and a lack of continuity of the variational problem. This paper shows how to build a variational equation for the efficient construction of Jacobians along trajectories of the delayed nonsmooth system. Trajectories of a piecewise smooth system may encounter the so-called grazing events, where the trajectory approaches discontinuity surfaces in the state space in a non-transversal manner. For these events we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrices computed by the algorithm converge with an order consistent with the order of the numerical integration method. Finally, we demonstrate the proposed method for a periodically forced impacting oscillator under a time-delayed feedback control, which exhibits grazing and crossing of the impact surface.