Stochastic stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises.

*(English)*Zbl 1331.93216Summary: The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.

##### MSC:

93E15 | Stochastic stability in control theory |

70K60 | General perturbation schemes for nonlinear problems in mechanics |

37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |

37M05 | Simulation of dynamical systems |

60H40 | White noise theory |

37H10 | Generation, random and stochastic difference and differential equations |

##### Keywords:

quasi-partially integrable Hamiltonian system; combined Gaussian and Poisson white noise excitations; asymptotic Lyapunov stability with probability one; the largest Lyapunov exponent
PDF
BibTeX
XML
Cite

\textit{W. Liu} et al., Nonlinear Dyn. 77, No. 4, 1721--1735 (2014; Zbl 1331.93216)

Full Text:
DOI

##### References:

[1] | 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 |

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

[3] | Grigoriu, M, Lyapunov exponents for nonlinear systems with Poisson white noise, Phys. Lett. A, 217, 258-262, (1996) · Zbl 0972.37520 |

[4] | Zhu, WQ, Lyapunov exponents and stochastic stability of quasi-non-integrable Hamiltonian systems, Int. J. Non-Linear Mech., 21, 569-579, (2004) · Zbl 1348.70064 |

[5] | 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 |

[6] | Zhu, WQ; Huang, ZL, Lyapunov exponents and stochastic stability of quasi-integrable-Hamiltonian systems, J. Appl. Mech., 66, 211-217, (1992) |

[7] | Khasminskii, RZ, Sufficient and necessary conditions of almost sure asymptotic stability of a linear stochastic system, Theory Probab. Appl., 12, 144-147, (1967) |

[8] | Kozin, F; Zhang, ZY, On almost sure sample stability of nonlinear differential equations, Probab. Eng. Mech., 6, 92-95, (1991) |

[9] | Roberts, JB; Spanos, PD, Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Non-Linear Mech., 21, 111-134, (1986) · Zbl 0582.73077 |

[10] | Zhu, WQ, Stochastic averaging method in random vibration, J. Appl. Mech. Rev., 41, 189-199, (1988) |

[11] | Zhu, WQ, Recent developments and applications of stochastic averaging method in random vibration, Appl. Mech. Rev., 49, s72-s80, (1996) · Zbl 1377.41019 |

[12] | Ariaratnam, ST; Tam, DSF, Lyapunov exponents and stochastic stability of coupled linear system, Probab. Eng. Mech., 6, 151-156, (1991) |

[13] | Ariaratnam, ST; Xie, WC, Lyapunov exponents and stochastic stability of coupled linear system under real noise excitation, J. Appl. Mech., 59, 664-673, (1992) · Zbl 0766.70017 |

[14] | Xu, Y; Wang, XY; Zhang, HQ; Xu, W, Stochastic stability for nonlinear systems deriven by L\(\acute{\text{ e }}\)vy noise, Nonlinear Dyn., 68, 7-15, (2012) · Zbl 1243.93126 |

[15] | Wojtkiwicz, SF; Johnson, EA; Bergman, LA; Grigoriu, M; Spencer, BF, Response of stochastic dynamics systems deriven by additive Gaussian and Poisson white noise: solution of a forward generalized Kolmogorov equation by a spectral finite difference method, Comput. Methods Appl. Mech. Eng., 168, 73-89, (1999) · Zbl 0956.70003 |

[16] | Zhu, HT; Er, GK; Iu, VP; Kou, KP, Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises, J. Sound Vib., 330, 2900-2909, (2011) |

[17] | Jia, W.T., Zhu, W.Q., 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) · Zbl 0766.70017 |

[18] | 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, J. Appl. Mech., 81, 014009, (2014) |

[19] | 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 |

[20] | 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 |

[21] | 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) |

[22] | 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) |

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

[24] | Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1993) |

[25] | Di Paola, M; Vasta, M, Stochastic integro-differential and differential equations of non-linear systems excited by parametric Poisson pulses, Int. J. Non-Linear Mech., 32, 855-862, (1997) · Zbl 0891.70014 |

[26] | Hanson, F.B.: Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, ZND Computation. SIAM, Philadelphia (2007) · Zbl 1145.60003 |

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

[28] | Khasminskii, RZ, On the averaging principle for stochastic \(It\hat{o}\) equation, Kibernetika, 4, 260-279, (1968) |

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

[30] | Zeng, Y; Zhu, WQ, Stochastic averaging of quasi-nonintegrable-Hamiltonian systems under Poisson white noise excitation, J. Appl. Mech., 78, 021002, (2011) |

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.