Extreme amplitudes of a periodically forced Duffing oscillator

A novel development in the theory of deterministic dynamical systems is the application of extreme value statistics. The idea is to evaluate a scalar observable along an evolution of the system. Under appropriate conditions the large values of this time series can be interpreted as realizations of a Generalized Extreme Value (GEV) distribution. The purpose of this paper is to illustrate the methodology with a time series of amplitudes of a periodically forced Duffing oscillator. The parameters of the GEV distribution are estimated by means of a block maximum method. It will be shown that a numerical estimation of these parameters can pose serious problems due to the fractal nature of the attractor of the system.