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Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems Abstract A new procedure is developed to study the stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Acta Mechanica Sinica" Springer Journals

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

Chunbiao, Gan
"Acta Mechanica Sinica" , Volume 20 (5): 9 – Oct 1, 2004
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References (18)

Publisher
Springer Journals
Copyright
2004 Chinese Society of Theoretical and Applied Mechanics
ISSN
0567-7718
eISSN
1614-3116
DOI
10.1007/BF02484279
Publisher site
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Abstract

Abstract A new procedure is developed to study the stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.

Journal

"Acta Mechanica Sinica" – Springer Journals

Published: Oct 1, 2004

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