Applications of the blow-up technique in singularly perturbed chemical kinetics - Miao, Zhouqian (The University of Edinburgh, 2019)
This thesis addresses the geometric analysis of traveling front propagation in singularly perturbed dynamical systems. The study of front propagation in reaction-diffusion systems has received a significant amount of attention in the past few decades. Frequently, of principal interest is the propagation speed of front solutions that connect various equilibrium states in these systems. Meanwhile, the geometric approach for normally hyperbolic problem is developed completely, and based on dynamical system theory. However, in the degenerate case, where the hyperbolicity is lost, we may consider the blow-up technique, which is also known as geometric desingularisation, to resolve the nonhyperbolic parts. We start with a two-component reaction-diffusion model with a small cut-off, which is a sigmoidal type of the FitzHugh-Nagumo system with Tonnelier-Gerstner kinetics. We first discuss the basic properties of the model without a cut-off, and we find two feasible cut-off systems for two components. We aim to construct a heteroclinic orbit connecting the non-zero equilibrium to the equilibrium at the origin for the cut-off system. However, the origin becomes degenerate due to the cut-off term. Hence, we apply the blow-up technique, which can resolve the degeneracy at the origin and regularize the dynamics in its neighborhood, where we can use standard dynamical system theory. We perform a formal linearisation and derive a second-order normal form in the blown-up dynamics to obtain the corresponding speed relation, which implies the existence of the heteroclinic orbit. We present the two blow-up patching approaches, numerical simulations and numerical comparison of the obtained results. We also discuss how the cut-off threshold is involved in the global geometry and the effect on the related propagating front speed and discontinuity position. The second main topic of the thesis is a geometric analysis of a reformulated singularly perturbed problem, based on the Martiel-Goldbeter model of a cyclic AMP (cAMP) signaling system, which models the propagation of cAMP signals during the aggregation of the amoeboid microorganism Dictyostelium discoideum. The mechanism is based on desensitisation of the cAMP receptor to extracellular cAMP. We explore the oscillatory dynamics of the reduced two-variable system without diffusion, which can be considered as the core mechanism in the cAMP signaling system, allowing for a phase plane analysis of oscillations due to the simplicity of the governing equations. There are two small parameters, which manifest very differently: while one parameter is a \conventional" singular perturbation parameter which reflects the separation of scales between the slow variable and the fast variable, the other parameter induces a different type of singular perturbation which is reflected by the non-uniformity of the limit. Our resolution, which introduces the blow-up technique to construct a family of periodic (relaxation-type) orbits for the singularly perturbed problem, uncovers a novel singular structure and improves our understanding of the corresponding oscillatory dynamics.
Keywords: front propagation; blow-up technique