On The Exit Problems In Nonlinear Dynamical Systems Driven By Random Perturbations

The concept of exit problem focuses on the random dynamical behaviors for the nonlinear dynamical systems,which are far from equilibrium states,in the limit of weak noise perturbations.It has already been known that,no matter how weak the noise is and no matter whether the dynamical systems are stable with probability 1 or not,a random dynamical system can always exit from the initial stationary state into another different stationary state,and the phenomena of noiseinduced transitions between different stationary states are called exit phenomena.Since random perturbations are ubiquitous in the real-life environment,the exit problems are widely involved in many disciplines and engineerings,like chemistry,biology,quantum physics,aerospace engineering,vehicle engineering,civil engineering and so on.Chaos is one of the most complicated phenomena in nonlinear dynamical systems.A dynamical system is said to be chaotic if it possesses three properties,i.e,the sensitive dependance on initial condition,the topologicaaly transitive and its periodic orbits being dense in the phase space.The peculiarity and the complexity of the exit phenomena in a chaotic system perturbed by random excitations have been a cause for general concern,and the relevant research results have important effects on the research for the exit behaviors of the general random nonlinear dynamical systems.Furthermore,since chaotic phenomena are widely found in mechanical,electrical,meteorological,biological and many other real-life systems,the study of the random behaviors in chaotic systems perturbed by random noises has a great significance in practical applications.In this dissertation,the exit behaviors of different nonlinear systems under different kinds of noises are studied.And the concepts of quasi-potential,mean first passage time(MFPT),most probable escape path(MPEP)are used to characterize the effects of chaotic attractors and chaotic saddles on the exit behaviors quantitatively.The primary contents and academic contributions of this dissertation are organized as follows:At first,a strongly nonlinear system with a bistable potential energy function driven by Gaussian white noise with and without periodic excitations is introduced respectively,for which the exit problems are investigated.Based on the WKB(Wenzel-Kramers-Brillouin)approximation,the singular perturbation method and the characteristic ray method,the FPK(Fokker-PlankKolmogorov)equation,which is a nonlinear partial differential equation of second order,is solved by transforming this equation into a group of ordinary differential equations on each characteristic ray.Then,the MFPT and MPEP of stochastic dynamical systems are obtained,and the results are verified by the use of Monte Carlo simulations and the concept of prehistory probability density.In the process of analysis,the singularities of the patterns of escape paths are found.And the positions of these singularities,the caustics and cusps,are determined by the analytical method.The relationship between the patterns of escape paths and MPEP is also analyzed.Based on the methods described above,the random behaviors for more complicated chaotic dynamical systems that are perturbed by weak noises are analyzed.In the Chapter 3,for a smooth and strongly nonlinear chaotic system,which is with quadratic and cubic terms and a periodic force,the global phase diagrams on a Poincaré section are derived firstly by use of the generalized cell-mapping and digraph(GCMD)method.Based on these,the finer fractal structures of a chaotic attractor and a chaotic saddle with respective parameters are investigated in the global phase diagrams.After a weak noise is introduced into this system,both the MFPT and MPEP are obtained respectively by the analytical methods given in Chapter 2.In addition,these results are verified by the numerical simulations and the analog electric circuit experiments.Finally,the influence of the chaotic attractors and chaotic saddles in the exit behaviors is analyzed based on the obtained results.For the chaotic saddles,there is a difficulty that they are hardly observed by the regular numerical methods because of their unattractive property;however,it is found that many unexpected complex phenomena of random dynamical systems are induced by the interaction between chaotic saddles and noises.In order to find this kind of the complex random dynamical behaviors,in Chapter 4,a piecewise linear system driven by Guassian white noise and a periodic force,which is a common model in engineering,is investigated.The bifurcation process for producing the chaotic saddle is observed by the use of GCMD,and with the perturbation of weak noises,the MFPT and MPEP for different parameters are both derived.Based on these results,the patterns of escapes in a nonlinear system with chaotic saddles are investigated,and the influence of the chaotic saddles on the reliability is also analyzed.Besides,the van der Pol transformation and stochastic averaging method are also applied to solve the exit problems of this system.And the obtained results for MFPT demonstrate that the period-3 cycles play an important roles in the exit behaviors through the chaotic saddle.In the Chapter 5,a Duffing oscillator forced by harmonic and real noise excitations is studied.The real noise used here is assumed ergodic and is defined as an integrable scalar function of an ndimensional Ornstein-Uhlenbeck(O-U)vector process,which is obtained by filtering a Gaussian white noise through a linear filter of first order.The spectrum representations of both the FokkerPlanck operator and its adjoint operator,and the asymptotic perturbation method are applied to obtain the FPK equation governing stationary probability density function.Due to these methods,the strong mixing condition and the detailed balance condition for the noise excitations are not involved here,so the exit problems induced by narrow band real noises can be considered in this dissertation.