Classical nonlinear response of a chaotic system: collective resonances

We consider the classical response in a chaotic system. In contrast to behavior in integrable or almost integrable systems, the nonlinear classical response in a chaotic system vanishes at long times. The response also reveals certain features of collective resonances which do not correspond to any periodic classical trajectories. The convergence of the response is shown to hold due to the exponential time dependence of the stability matrix. The growing exponentials corresponding to strong instability do not inhibit the convergence. We calculate both linear and second-order response in one of the simplest chaotic systems: free classical motion on a surface of constant negative curvature. We demonstrate the relevance of the model for applications to spectroscopic experiments.