Bifurcation trees of periodic motions to chaos in a parametric Duffing oscillator

In this paper, bifurcation trees of periodic motions to chaos in a damped, parametric Duffing oscillator are investigated. From the semi-analytic method, differential equations of nonlinear dynamical systems are discretized first to obtain implicit mappings. Following the implicit mapping structures, periodic nodes of periodic motions are computed. The bifurcation trees of period-1 to period-4 motions are presented to demonstrate the routes of period-1 motions to chaos, and the corresponding stability and bifurcation are determined by eigenvalue analysis. For a better understanding of nonlinear behaviors of periodic motions in a parametric Duffing oscillator, harmonic frequency–amplitude characteristics of periodic motions are presented. From the analytical predictions, numerical simulations are performed. The trajectory, time-histories of displacement and velocity, harmonic amplitudes and phases of period-1 to period-4 motions are presented. Based on comparison of numerical and analytical results, determined is how many harmonic terms should be included in finite Fourier series, which help one select harmonic terms in analytical solutions and engineering application.