Finite-time generalized high-order Lyapunov exponents for kicked double rotor

The ordinary Lyapunov exponents spectrum describes the average exponential expansion/shrinkage rates of the axis of an infinitesimal ball around a trajectory under the temporal evolution of a dynamical system. These exponents are given by the linearization of the ruling equations. Due to intrinsic nonlinearities present in models that present chaotic dynamics, nonlinear effects, swept off by the linearization, can be crucial in elucidating details of the temporal evolution of such systems. Moreover, since Lyapunov exponents are dynamic invariants computed as an average over an ergodic trajectory, they are "blind" about local/finite-time fluctuations present in typical chaotic dynamical systems. We present a detailed analysis of the finite-time fluctuations of the generalized high-order Lyapunov exponents for a physical system composed of a periodically kicked double rotor. We focus in its chaotic regime and in the transition from chaos to hyper-chaos as the intensity of the kicks is increased. Generalized high-order Lyapunov exponents are given by the analysis of high-order derivatives of the dynamical equations, which define linear mappings and their effects over the Lyapunov vectors are studied in a similar manner done for ordinary Lyapunov exponents. We study the temporal fluctuations of these high-order Lyapunov exponents for finite-time trajectories and relate their properties with those observed in the chaos / hiper-chaos transition.