Lyapunov function method for analyzing stability of quasi-Hamiltonian systems under combined Gaussian and Poisson white noise excitations.

*(English)*Zbl 1348.37023Summary: The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi-Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied by using Lyapunov function method. According to the integrability and resonance, quasi-Hamiltonian systems can be divided into five classes, namely quasi-non-integrable, quasi-completely integrable and non-resonant, quasi-completely integrable and resonant, quasi-partially integrable and non-resonant, and quasi-partially integrable and resonant. Lyapunov functions for these five classes of systems are constructed. The derivatives for these Lyapunov functions with respect to time are obtained by using the stochastic averaging method. The approximately sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian system under parametric excitations of combined Gaussian and Poisson white noises is determined based on a theorem due to Khasminskii. Four examples are given to illustrate the application and efficiency of the proposed method. And the results are compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent method.

##### MSC:

37B25 | Stability of topological dynamical systems |

70H08 | Nearly integrable Hamiltonian systems, KAM theory |

70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |

70K65 | Averaging of perturbations for nonlinear problems in mechanics |

##### Keywords:

quasi-Hamiltonian system; combined Gaussian and Poisson white noise excitations; stochastic stability; Lyapunov function; stochastic averaging
PDF
BibTeX
XML
Cite

\textit{W. Liu} and \textit{W. Zhu}, Nonlinear Dyn. 81, No. 4, 1879--1893 (2015; Zbl 1348.37023)

Full Text:
DOI

##### References:

[1] | Zhu, WQ, Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems, Int. J. Non-Linear Mech., 39, 569-579, (2004) · Zbl 1348.70064 |

[2] | Zhu, WQ; Huang, ZL, Lyapunov exponent and stochastic stability of quasi-integrable-Hamiltonian systems, ASME J. Appl. Mech., 66, 211-217, (1999) |

[3] | Zhu, WQ; Huang, ZL; Suzuki, Y, Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems, Int. J. Non-Linear Mech., 37, 419-437, (2002) · Zbl 1346.70013 |

[4] | Ling, Q; Jin, XL; Li, HF; Huang, ZL, Lyapunov function construction for nonlinear stochastic dynamical systems, Nonlinear Dyn., 72, 853-864, (2013) · Zbl 1284.93207 |

[5] | Huang, ZL; Jin, XL; Zhu, WQ, Lyapunov functions for quasi-Hamiltonian systems, Probab. Eng. Mech., 24, 374-381, (2009) |

[6] | Liu, WY; Zhu, WQ; Xu, W, Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises, Probab. Eng. Mech., 32, 39-47, (2013) |

[7] | Liu, WY; Zhu, WQ; Jia, WT, Stochastic stability of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises, Int. J. Non-Linear Mech., 58, 191-198, (2014) |

[8] | Liu, WY; Zhu, WQ; Jia, WT; Gu, XD, Stochastic stability of quasi partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises, Nonlinear Dyn., 77, 1721-1735, (2014) · Zbl 1331.93216 |

[9] | Liu, WY; Zhu, WQ, Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises, Int. J. Non-Linear Mech., 67, 52-62, (2014) |

[10] | Oseledec, VI, A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems, Trans. Mosc. Math. Soc., 19, 197-231, (1968) · Zbl 0236.93034 |

[11] | Zhu, WQ; Yang, YQ, Stochastic averaging of quasi-non-integrable-Hamiltonian systems, ASME J. Appl. Mech., 64, 157-164, (1997) · Zbl 0902.70013 |

[12] | Zeng, Y; Zhu, WQ, Stochastic averaging of quasi-nonintegrable-Hamiltonian systems under Poisson white noise excitation, ASME J. Appl. Mech., 78, 021002-1-021002-11, (2010) |

[13] | Jia, WT; Zhu, WQ; Xu, Y, Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations, Int. J. Non-Linear Mech., 51, 45-53, (2013) |

[14] | Khasminskii, R.Z.: Stochastic Stability of Differential Equation. Sijthoff & Noordhoff, Alphen aan den Rijn (1980) |

[15] | Paola, M; Falsone, G, \(It\hat{o}\) and Stratonovich integrals for delta-correlated processes, Probab. Eng. Mech., 8, 197-208, (1993) |

[16] | Paola, M; Falsone, G, Stochastic dynamics of nonlinear systems driven by non-normal delta-correlated process, ASME J. Appl. Mech., 60, 141-148, (1993) · Zbl 0778.70022 |

[17] | Zhu, WQ; Huang, ZL, Stochastic averaging of quasi-integrable Hamiltonian systems, ASME J. Appl. Mech., 64, 975-984, (1997) · Zbl 0918.70009 |

[18] | Jia, WT; Zhu, WQ, Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations, Nonlinear Dyn., 76, 1271-1289, (2014) · Zbl 1306.70010 |

[19] | Zhu, WQ; Huang, ZL; Suzuki, Y, Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems, Int. J. Non-Linear Mech., 37, 419-437, (2002) · Zbl 1346.70013 |

[20] | Jia, WT; Zhu, WQ, Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations, J. Phys. A, 398, 125-144, (2014) · Zbl 1395.82112 |

[21] | Jia, WT; Zhu, WQ; Xu, Y; Liu, WY, Stochastic averaging of quasi-integrable and resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations, ASME J. Appl. Mech., 81, 041009, (2014) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.