Asymptotic Studies On Piecewise-Smooth Singularly Perturbed Dynamical Systems With Multiple Roots Of The Degenerate Equations

For piecewise-smooth dynamical systems with small parameters,there have been a lot of researches in the case of isolated roots on both sides of discontinuous curve or surface.As a mathematical model,this kind of problem has been widely used in the fields of mechanics,superconducting science and life science.However,with the development of research interests,piecewise-smooth dynamical systems with multiple roots of the degenerate equations have aroused great interest of a great many mathematicians.In this case,constructing the asymptotic solution is very difficult,because the classical boundary layer function method and Fenichel's three theorems are no longer applicable.In this thesis,by using asymptotic method,geometric theory and numerical method,we mainly study some piecewise-smooth dynamical systems with multiple roots of the degenerate equations and a class of singularly perturbed boundary value problems with intersecting roots in the case when one of the degenerate solutions is multiple roots.This thesis aims to construct multi-scale smooth asymptotic solutions.The obtained results variably generalize and improve the previous results found in literature.The dissertation is divided into seven chapters.The first chapter introduces the research background and latest progress of piecewisesmooth singularly perturbed dynamical systems with multiple roots of the degenerate equations,singularly perturbed problems with turning points and asymptotic-numerical methods,then briefly states the main research work and main results of this thesis.And the last section is the summary of the research results of this dissertation and the plan of the follow-up work.In Chapter 2,the multizonal internal layer of the stationary solution of the piecewisesmooth reaction-diffusion equation with double root of the degenerate equation is studied.After decomposing into two classical boundary value problems,some conditions to guarantee the existence of a smooth solution is given and the asymptotic solution is constructed by modified boundary layer method.Here the internal layer can be divided into six regions,and the internal layer function experiences a transition from algebraic decay to exponential decay.Then,based on the existence theorem of pure boundary value problem,it is proved that the original problem has multizonal internal layer solutions by using the matching method.Finally,the results are verified by a numerical simulation.In Chapter 3,the asymptotic behavior of the stationary solution of the reactiondiffusion-convection equation is proposed and analyzed.Firstly,the existence of solutions of pure boundary value problems on both sides of discontinuous lines is discussed and their asymptotic solutions are constructed.More importantly,it is concluded that boundary layer function changes from algebraic attenuation to exponential attenuation by using the comparison principle.Then by analysis,the internal layer is also divided into six regions.And a numerical solution for the leading term of boundary layer function is obtained by Laguerre pseudospectral method.Finally,after smooth matching these two classical problems on the discontinuous line,the uniformly effective smooth solution on the whole interval is obtained,and then the existence of the solution is proved,and the remainder estimation is given.The results are verified by numerical examples.In Chapter 4,the asymptotic analysis of two-point boundary value problems for two second-order ordinary differential equations with discontinuous right hands with multiple roots of the degenerate equation is carried out.In this case,the small parameters before the highest order terms of these equations have different powers.Not only the asymptotic solution is constructed by using the modified boundary layer function method,but also the internal layer function has a transition from algebraic decay to exponential decay and the internal layer is decomposed into eight regions.Based on the contrast structure theory of classical boundary value problems and the matching method,the existence of smooth solutions and the uniform validity of asymptotic solutions are proved,and the error estimation is carried out.In Chapter 5,for weakly nonlinear dynamical systems with isolated roots and semilinear singularly perturbed boundary value problems with triple roots,the internal layer with the same multiplicity of degenerate roots of the degenerate equations in the case when the discontinuous curve is a more general monotone curve is studied.Using the boundary layer function method and the modified boundary layer function method,the multi-scale asymptotic representation in two regions divided by the discontinuous curve is constructed respectively.And they are connected smoothly on the discontinuous curve,and the existence of internal layer solutions are proved.In particular,it is necessary to prove the existence of solutions to continuous semilinear dynamical systems with triple roots of the degenerate equations.In Chapter 6,the asymptotic behavior of the stationary solution of singularly perturbed one-dimensional reaction-diffusion equation with different multiplicities of degenerate roots on both sides of the discontinuous curve is discussed.One is in the case when the degenerate equations on both sides of the discontinuous curve have a double root and an isolated root.The other is that degenerate equations have a triple root and a double root,respectively.Not only the formal asymptotic solution is constructed by the modified boundary layer function method,but also the asymptotic expansion of the point passing through the discontinuous curve is determined by some techniques.Moreover,the existence of a uniformly effective smooth solution in the whole interval is proved after smooth matching,and the remainder estimation of the error is given.Finally,some numerical examples are given to verify the theoretical results.In Chapter 7,we propose and deeply study a class of singularly perturbed systems in the case when the degenerate critical manifold is non-hyperbolic everywhere.This is a problem that degenerate equations have intersecting roots.In particular,the degenerate equation has a double root and an isolated root.Based on the traditional method in the case of isolated roots,the non-homogeneous blowup technique and the method of phase plane analysis are used to analyze the dynamic behavior of the solution near the critical manifold(especially the turning point).The zero order asymptotic expansion of the solution is given by using the regularization method.And the asymptotic behavior near the boundary layer is discussed by combining the boundary layer function method with the modified boundary layer function method.Moreover,the asymptotic differential inequality method is used to prove the existence of the solution and give the error estimation.An example is presented to illustate the obtained results.