Towards Infinite Bifurcation Trees of Period-1 Motions to Chaos in a Time-delayed, Twin-well Duffing Oscillator

Abstract

In this paper, bifurcation trees of periodic motions to chaos in a periodically forced, time-delayed, twin-well Duffing oscillator are pre- dicted by a semi-analytical method. The twin-well Duffing oscilla tor is extensively used in physics and engineering. The bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is very significant for determine motion complexity. Thus, the bi furcation trees for periodic motions to chaos in such a time-delayed, twin-well Duffing oscillator are obtained analytically. From the fi nite discrete Fourier series, harmonic frequency-amplitude character istics for period-1 to period-4 motions are analyzed. The stability and bifurcation behaviors of the time-delayed Duffing oscillator are different from the non-time-delayed Duffing oscillator. From the an alytical prediction, numerical illustrations of periodic motions in the time-delayed, twin-well Duffing oscillator are completed. The com plexity of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. As a slowly varying excitation becomes very slow, the excitation amplitude will approach infinity for the infinite bifurcation trees of period-1 motion to chaos. Thus infinite bifurca tion trees of period-1 motion to chaos can be obtained.