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A Smooth and Discontinuous OscillatorResonancesResonance

A Smooth and Discontinuous Oscillator: ResonancesResonance [Attention is now paid to the resonance behaviour of the discontinuous case of the oscillator. Although the method and the results presented here are for the discontinuous case, they are also valid for the smooth dynamics. To clarify the resonant phenomena, we introduce the concepts of generalized winding number and the generalized canonical generating functions to derive the complicated resonant dynamics for the discontinuous system. The time dependent Hamiltonian for the discontinuous system enables us to understand such a resonant motion by a composed winding number consisting of multiple phases, each of which is confined within one of the separate resonant area, satisfying the regular resonant condition. The complex resonant structures of stochastic webs or chaotic sea with stochastic trajectories entering different resonant layers of resonant island chains observed are considered as the combination of the separated linear system according to the composed winding behaviour. This might suggest further investigations into resonant synchronization and mechanisms of vibrational power flow transfer from nonlinear smooth to discontinuous dynamical systems.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Smooth and Discontinuous OscillatorResonancesResonance

Part of the Springer Tracts in Mechanical Engineering Book Series
Cao, Qingjie; Léger, Alain
Springer Journals — Sep 28, 2016
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8 pages

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Publisher
Springer Berlin Heidelberg
Copyright
© Springer-Verlag Berlin Heidelberg 2017
ISBN
978-3-662-53092-4
Pages
43 –51
DOI
10.1007/978-3-662-53094-8_4
Publisher site
See Chapter on Publisher Site

Abstract

[Attention is now paid to the resonance behaviour of the discontinuous case of the oscillator. Although the method and the results presented here are for the discontinuous case, they are also valid for the smooth dynamics. To clarify the resonant phenomena, we introduce the concepts of generalized winding number and the generalized canonical generating functions to derive the complicated resonant dynamics for the discontinuous system. The time dependent Hamiltonian for the discontinuous system enables us to understand such a resonant motion by a composed winding number consisting of multiple phases, each of which is confined within one of the separate resonant area, satisfying the regular resonant condition. The complex resonant structures of stochastic webs or chaotic sea with stochastic trajectories entering different resonant layers of resonant island chains observed are considered as the combination of the separated linear system according to the composed winding behaviour. This might suggest further investigations into resonant synchronization and mechanisms of vibrational power flow transfer from nonlinear smooth to discontinuous dynamical systems.]

Published: Sep 28, 2016

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