On complex periodic motions and bifurcations in a periodically forced, damped, hardening Duffing oscillator. (English) Zbl 1355.34060
Summary: In this paper, analytically predicted are complex periodic motions in the periodically forced, damped, hardening Duffing oscillator through discrete implicit maps of the corresponding differential equations. Bifurcation trees of periodic motions to chaos in such a hardening Duffing oscillator are obtained. The stability and bifurcation analysis of periodic motion in the bifurcation trees is carried out by eigenvalue analysis. The solutions of all discrete nodes of periodic motions are computed by the mapping structures of discrete implicit mapping. The frequency-amplitude characteristics of periodic motions are computed that are based on the discrete Fourier series. Thus, the bifurcation trees of periodic motions are also presented through frequency-amplitude curves. Finally, based on the analytical predictions, the initial conditions of periodic motions are selected, and numerical simulations of periodic motions are carried out for comparison of numerical and analytical predictions. The harmonic amplitude spectrums are also given for the approximate analytical expressions of periodic motions, which can also be used for comparison with experimental measurement. This study will give a better understanding of complex periodic motions in the hardening Duffing oscillator.
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 37C75 | Stability theory for smooth dynamical systems |
| 34C40 | Ordinary differential equations and systems on manifolds |
Keywords:
hardening Duffing oscillator; discrete implicit maps; analytical bifurcation trees; complex period-\(m\) motionsReferences:
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