Research On The Applications Of Geometric Singular Perturbation Theory

Geometric singular perturbation theory is a powerful tool in analyzing complex nonlinear systems with multiple time scales. The basic idea is that the dynamical behaviors of a singular perturbed system can be obtained by analyzing the corresponding limiting slow and limiting fast system. This dissertation aims to investigate the applications of geometric singular perturbation theory to several types of specific nonlinear models. The dissertation consists of six chapters. The main contents are as follows:In chapter 1, the development and applications of the geometric singular perturbation theory are introduced briefly, and the main contributions of this dissertation are outlined.In chapter 2, we give some preliminary tools which will be used in this dissertation.In chapter 3, we investigate stationary waves of non-isentropic compressible flows through an expanding-contracting nozzle. We firstly regard the thermal conduction coefficient as a singular small parameter and transform the problem of stationary waves into a singular perturbation problem. Then we give a classification of singular stationary waves by analyzing the dynamics of limiting slow and limiting fast systems. Finally, by making use of the geometric singular perturbation theory, we prove the persistence of singular stationary waves.In chapter 4, we study the existence of slowly modulated two-pulse solutions for a generalized Klausmeier-Gray-Scott model. By applying geometric singular perturbation theory combined with Melnikov method, we firstly present the phase space geometry associated with the pulse solutions. Then based on the geometry of the invariant manifolds, we give a formal geometric construction of the slowly modulated two-pulse solutions by employing geometric singular perturbation theory. Finally, we derive the parameter constraint conditions under which the model admits slowly modulated two-pulse solutions, and also determine the implicit differential equations for the wave speed for the left and right pulse wave.In chapter 5, we deal with the mixed-mode oscillation dynamics in an extended Bonhoeffer-van der Pol oscillator. We firstly obtain the dynamical behaviors of the corresponding layer problem and reduced problem by means of geometric singular perturbation analysis. By applying geometric singular perturbation theory combined with the theory of canard-induced mixed mode oscillations, we derive a parameter regime in which the model exhibits a stable periodic orbit of mixed-mode oscillation type.In chapter 6, we investigate the dynamical behaviors of a viscoelastic fluid model. We firstly investigate the small scale dynamics of the model by using geometric singular perturbation theory. By making coordinate scale transformation,the model is transformed into the corresponding large scale system. Finally, by applying the blow-up method combined with geometric singular perturbation theory, we successfully glue together the dynamics of the small scale and the large scale, and thus obtain the global dynamics of the model.