On Stochastic Stabilities And Stochastic Bifurcations For Nonlinear Real Noise Systems

Stability of random dynamical systems has always been one of the focuses of theoretical researches of random dynamics,which has extensive applications in aerospace engineering,marine engineering,vehicle engineering,industrial and civil construction engineering,defence engineering,etc.In the present thesis,the stochastic stabilities,characterized by the moment Lyapunov exponents,and the stochastic bifurcations of a co-dimension two-bifurcation system that is in a threedimensional center manifold,a binary airfoil,a single degree of freedom linear oscillator with a fractional order damping and a viscoelastic wallboard under the excitation of a Gaussian real noise are investigated respectively.The main contents as follows:Moment Lyapunov exponent of a co-dimension two-bifurcation system in a three-dimensional center manifold under the parametric excitation of an ergodic real noise,which is defined as an integrable scalar function of an n-dimensional O-U vector process,is investigated firstly.To make the model more general,the general conditions for the noise excitation,i.e.the strong mixing condition and the detailed balance condition are deleted.The spectrum representations of both the Fokker-Planck operator and its adjoint and the L.Arnold perturbation method are employed to evaluate the asymptotic expression of the finite p-th moment Lyapunov exponent.Furthermore,the rationality of this method is verified by comparison the asymptotic expression of the finite p-th moment Lyapunov exponent with the results of numerical simulation.Finally,the variations of the approximate p-th moment Lyapunov exponent with the system parameters and the effects of various noise parameters on stochastic stability are studied.Secondly,the random flutter of a binary airfoil subjected to turbulence is considered.The turbulent disturbance is emulated as an ergodic real noise so that the problem translates into the stochastic bifurcation and stochastic stability of a four-dimensional system.The asymptotic expressions of the finite p-th moment Lyapunov exponent are acquired by using the L.Arnold perturbation method and the spectrum representations of both the Fokker-Planck operator and its adjoint for the linear filtering system.Then a comparison with the numerical results proves its validity and an analysis of the effects of various system parameters on stochastic stability is conducted with recourse to the analytical results of the moment Lyapunov exponent.Chapter 4 is devoted to the stochastic stability of a single degree of freedom linear oscillator with a fractional order damping term excited by both a harmonic load and a real noise.By means of the polar coordinate transformation,the fractional order damping is approximated,leading to a transformed equation without fractional derivatives.Moreover,a similar procedure as above gives two important indicators of stochastic stability: the asymptotic expressions of the maximal Lyapunov exponent and the moment Lyapunov exponent.Based on that,a comparison and analysis of the influence of fractional order ? are carried out and it can be concluded that the inherent frequency has significant impact on stochastic stability due to the introduction of fractional derivatives.The random flutter of a viscoelastic plate,of which the viscoelasticity is described by a fractional Kelvin–Voigt constitutive relation,is discussed in Chapter 5.A direct application of piston theory provides the first two order models of the viscoelastic plate subjected to stochastic excitations and consequently a group of four-dimensional governing equations,given by stochastic differential equations,is acquired.Subsequent to the maximal Lyapunov exponent and the moment Lyapunov exponent gained respectively in both cases of resonance and non-resonance,stochastic stability of this system and how the introduction of the fractional Kelvin–Voigt constitutive relation affects it are discussed in detail.