Dynamics And Control Of Some Classes Of Non-linear Stochastic Systems

In Chapter 2, the generalized Hamiltonian systems (GHS) are classified as five groups based on their integrability and resonance, i.e., completely non-integrable, completely integrable and non-resonant, completely integrable and resonant, partially integrable and non-resonant, partially integrable and resonant. In Chapter 3, the functional form of the exact stationary solutions and the conditions for exact stationary solutions to exist for five groups of dissipated GHS under Gaussian white noise excitations are given. In Chapter 4, the equivalent nonlinear system methods for dissipated five classes of GHS under Gaussian white noise excitations are developed. Three equivalence criteria were proposed for finding the equivalent nonlinear system and their stationary solutions. They are minimization of the mean square deficiency in damping forces, minimization of the mean square deficiency in the energies dissipated by the damping forces, and equality of the expected time rates of the first integrals of the given system and its equivalent one. In Chapter 5, stochastic averaging methods for five classes of GHS under light damping and Gaussian white noise excitations are developed. The form and dimension of the averaged Ito stochastic differential equation depends upon the integrability and resonance of the associated GHS. The diffusion and drift coefficients of the averaged Ito equation are given in detail. The stochastic averaging methods for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations, under bounded noise excitations, and under stationary wide-band noise excitations, respectively, are also developed. The deterministic averaging method for completely integrable Hamiltonian system under light damping and weak harmonic excitations is the special case of the stochastic averaging method for dissipated completely integrable Hamiltonian system under combined harmonic and Gaussian white noise excitations. The developed stochastic averaging methods are used to predict the responses of single-degree-of-freedom vibro-impact system under light damping and stationary wide-band noise excitations, and the responses of multi-degree-of-freedom vibro-impact systems under light damping and Gaussian white noise excitations. In Chapter 6, the largest Lyapunov exponent and almost sure asymptotic stability of quasi non-integrable GHS under light damping and weak Gaussian white noise excitations are studied based on the stochastic averaging methods for quasi-non-integrable GHS and the definition of new norm in terms of square root of summation of Hamiltonian and Casimir functions. A new approach to the almost sure asymptotic stability of linear stochastic system is proposed by using the property that the largest Lyapunov exponent is independent of the positive-definite quadratic form in the definition of norm. The Lyapunov asymptotic stability with probability one of quasi Hamilton systems are studied by using Lyapunov function and the stochastic averaging methods for quasi Hamilton systems. The Lyapunov functions are the optimal linear combination of the first integrals of the associated completely integrable and non-resonant or partially integrable and non-resonant Hamiltonian systems. An approximate method for determing the stochastic Hopf bifurcation of quasi integrable Hamiltonian system is developed based on the stochastic averaging...