Research On Chaotic Dynamics Of Linear Hyperbolic Systems With Nonlinear Boundary Conditions

As a characterization of complexity of dynamical systems,chaos widely exists in nature.It is well known that the chaos theory in finite-dimensional dynamical systems has been well-developed.However,less is known about the theory of chaos in the systems governed by partial differential equations(PDEs).Compared with the chaos theory in finite-dimensional dynamical systems,the study of the chaos theory in systems of PDEs requires more sophisticated mathematical techniques.With the presence of nonlinearities in the system of PDE,the basic issues such as existence and uniqueness of solutions often fail to well settle,much less the determination of chaotic behavior.In this dissertation,the chaotic dynamics of the systems governed by second-order linear hyperbolic PDEs with nonlinear boundary conditions are studied extensively.By using of characteristics,the solutions of the systems are analytically expressed.Combined with some methods and the related theory of total variation and discrete dynamical systems,it is proved that the systems are chaotic in the sense of exponential growth of total variation with time.Numerical examples are demonstrated to illustrate the effectiveness of theoretical results.Chapter 1 gives a brief survey to the background of this dissertation,including the definitions,characteristics,research advance of chaos,some related concepts and results of chaos theory in the systems governed by PDEs,and the main works of this dissertation.Chapter 2 studies the chaotic dynamics of the initial-boundary value problem of the one-dimensional wave equation with a mixing transport term.It separately considers that the boundary condition at the right-end of the wave equation is a superlinear type and linear perturbation of such type,each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term,which implies that the interaction of the boundary condition and equation leads to the change of system energy.For each type of boundary condition,the existence of chaos is rigorously proved.Chapter 3 establishes rigorously mathematical theorems that guarantee the existence of chaos in the system of second-order linear hyperbolic PDE with constant coefficients.It separately considers the system with nonlinear implicit boundary conditions(IBCs)as well as those with such IBCs subjected to small perturbations,where IBCs here are difficult to explicitly express in general.Chapter 4 analyses the chaotic dynamics of the initial-boundary value problem of the second-order linear hyperbolic PDE with constant coefficients and nonlinear boundary condition(NBC).Compared with the boundary conditions in all previous related literatures,such NBC is more general.The interval of parameter of the system for the occurrence of chaos are precisely characterized.The chaotic results obtained here are further applied to superlinear boundary condition,polynomial boundary condition and telegraph equation,respectively.Chapter 5 discusses chaotic dynamics of the initial-boundary value problem of the second-order linear hyperbolic PDE with variable coefficients and both boundary conditions being general NBCs.Under certain conditions,the second-order linear hyperbolic PDE with variable coefficients can be factorized as a product of two first order operators by operator-factoring technique.The existence of chaos is further rigorously proved.