TY - JOUR
T1 - Resonant dynamics of a periodically driven noisy oscillator
AU - Choi, Seunggil
AU - Sri Namachchivaya, N.
AU - Onu, Kristjan
N1 - Funding Information:
The authors would like to acknowledge the support of the AFOSR under grant number FA9550-08-1-0206 and the National Science Foundation under grant number CMMI 07-58569 . Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/1
Y1 - 2011/1
N2 - A unified approach to study the dynamics of single degree of freedom systems excited by both weakly periodic and random additive perturbations is developed. The near-resonant motion of such systems is not well understood. This problem is studied with the aim of discovering a common geometric structure in the phase space, and determining the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. The solution of the reduced model can be approximated by a Markov process. Depending on the strength of the noise, the reduced Markov process takes its values on a line or on a graph with certain gluing conditions at the vertices of the graph. The reduced model provides a framework for computing standard statistical measures of dynamics and stability such as probability density functions. This work also explains a counter-intuitive phenomenon in stochastic resonance, in which a weak periodic forcing in a nonlinear system can be enhanced by the addition of external noise.
AB - A unified approach to study the dynamics of single degree of freedom systems excited by both weakly periodic and random additive perturbations is developed. The near-resonant motion of such systems is not well understood. This problem is studied with the aim of discovering a common geometric structure in the phase space, and determining the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. The solution of the reduced model can be approximated by a Markov process. Depending on the strength of the noise, the reduced Markov process takes its values on a line or on a graph with certain gluing conditions at the vertices of the graph. The reduced model provides a framework for computing standard statistical measures of dynamics and stability such as probability density functions. This work also explains a counter-intuitive phenomenon in stochastic resonance, in which a weak periodic forcing in a nonlinear system can be enhanced by the addition of external noise.
KW - Hamiltonian dynamics
KW - Nonlinear resonance
KW - Stochastic averaging
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U2 - 10.1016/j.probengmech.2010.08.001
DO - 10.1016/j.probengmech.2010.08.001
M3 - Article
AN - SCOPUS:77957753539
SN - 0266-8920
VL - 26
SP - 109
EP - 118
JO - Probabilistic Engineering Mechanics
JF - Probabilistic Engineering Mechanics
IS - 1
ER -