Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises
A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant rel...
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| Published in | International journal of non-linear mechanics Vol. 67; pp. 52 - 62 |
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| Format | Journal Article |
| Language | English |
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01.12.2014
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| Abstract | A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant relations, the averaged Ito^ stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii׳s procedure to the averaged Ito^ SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation.
•The case of resonance with α resonant relations is considered.•The excitations are the combined Gaussian and Poisson white noises excitations.•Stochastic averaging method is used to derive the averaged equations.•The expression for the largest Lyapunov exponent of the system is formulated.•Theoretical results agree well with those from Monte Carlo simulation. |
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| AbstractList | A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant relations, the averaged Ito^ stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii׳s procedure to the averaged Ito^ SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation.
•The case of resonance with α resonant relations is considered.•The excitations are the combined Gaussian and Poisson white noises excitations.•Stochastic averaging method is used to derive the averaged equations.•The expression for the largest Lyapunov exponent of the system is formulated.•Theoretical results agree well with those from Monte Carlo simulation. A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with alpha resonant relations, the averaged It stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii's procedure to the averaged Ito SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation. |
| Author | Zhu, Weiqiu Liu, Weiyan |
| Author_xml | – sequence: 1 givenname: Weiyan surname: Liu fullname: Liu, Weiyan organization: Department of Applied Mathematics, Northwestern Polytechnical University, Xi׳an 710072, China – sequence: 2 givenname: Weiqiu surname: Zhu fullname: Zhu, Weiqiu email: wqzhu@yahoo.com organization: Department of Applied Mathematics, Northwestern Polytechnical University, Xi׳an 710072, China |
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| Cites_doi | 10.1016/0021-8928(82)90131-9 10.1016/S0020-7462(01)00018-X 10.1115/1.4025141 10.1007/s11071-013-0757-3 10.1016/S0020-7462(00)00006-8 10.1016/j.ijnonlinmec.2013.09.010 10.1002/eqe.4290190408 10.1016/j.probengmech.2008.11.001 10.1016/0266-8920(91)90022-V 10.1007/s11071-014-1413-2 10.1016/j.probengmech.2012.12.009 10.1016/j.ijnonlinmec.2012.12.003 10.1016/S0020-7462(02)00223-8 10.1016/j.physd.2011.06.001 10.1016/S0020-7462(96)00081-9 10.1115/1.2893775 10.1115/1.2900736 10.1016/S0020-7462(99)00047-5 10.1137/1112019 10.1006/jsvi.2000.2951 10.1115/1.2193137 10.1016/0266-8920(93)90015-N 10.1007/978-94-009-9121-7 10.1115/1.2948382 10.1115/1.2789148 10.1115/1.2789009 |
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| Keywords | Stochastic stability Quasi-integrable and resonant Hamiltonian system Stochastic averaging Combined Gaussian and Poisson white noise excitations Lyapunov exponent |
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| SubjectTerms | Combined Gaussian and Poisson white noise excitations Computer simulation Excitation Gaussian Lyapunov exponent Lyapunov exponents Mathematical analysis Quasi-integrable and resonant Hamiltonian system Stability Stochastic averaging Stochastic stability Stochasticity White noise |
| Title | Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises |
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