On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator

Abstract

In this paper, bifurcation trees of period-3 motions to chaos in a periodically forced, time-delayed, twin-well Duffing oscillator are investigated. Such period-3 motions and corresponding bifurcation trees may or may not be embedded in chaos in the bifurcation trees of period-1 motions to chaos. The bifurcation trees for period-3 motions to chaos in such a time-delayed, twin-well Duffing oscillator are obtained analytically. From the finite discrete Fourier series, harmonic frequency-amplitude characteristics for period-3 to period-6 motions are analyzed. From the analytical prediction, numerical illustrations of period-3 and period-6 motions in the time-delayed, twin-well Duffing oscillator are completed. The complexity of period-3 motions to chaos in nonlinear dynamical systems is strongly dependent on the distributions and quantity levels of harmonic amplitudes. For a very slowly varying excitation, the infinite bifurcation trees of period-3 motions to chaos can be achieved once the excitation amplitude approaches to infinity.