PERIODICITY OF AVERAGED HISTORIES OF CHAOTIC OSCILLATORS

This study is dedicated to demonstrate the periodicities embedded in the averaged responses of chaotic systems with periodic excitations. Recent studies in the field of non-linear oscillations often found random-like responses for some deterministic non-linear systems with periodic excitations, which were then named “chaotic systems”. However, in this study, by discretizing the initial conditions on a chosen domain and averaging the corresponding responses, the averaged response can be calculated for the chaotic motions of Duffing, van der Pol and piecewise linear systems. These averaged responses exhibit near-periodicities with primary frequency components at excitation frequency, odd multiples or half multiples of excitation frequency. It is also found that this periodicity becomes more evident as the number of discretized initial conditions over a fixed domain. These results were obtained and validated by simulations.