Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

• Published in: International journal of non-linear mechanics Vol. 37; no. 3; pp. 419 - 437
• Main Authors: Zhu, W.Q., Huang, Z.L., Suzuki, Y.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.04.2002; Elsevier Science
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Lyapunov exponent; Mathematical analysis; Mathematics; Non-linear system; Partial differential equations; Physics; Response; Sciences and techniques of general use; Solid mechanics; Stochastic averaging; Stochastic excitation; Stochastic stability; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves; Response; Stochastic stability; Non-linear system; Stochastic excitation; Stochastic averaging; Lyapunov exponent; Averaging method; Itô equation; Equation of motion; Random excitation; Integrable system; Numerical method; Stochastic integral; Non linear oscillator; First integral; Fokker Planck equation; Hamiltonian mechanics; Hamiltonian system; Resonance; Random vibration
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(01)00018-X

An n degree-of-freedom Hamiltonian system with r (1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker–Planck–Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.