Asymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction

Асимптотика, устойчивость и область притяжения периодического решения сингулярно возмущённой параболической задачи с двукратным корнем вырожденного уравнения

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in a small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.Для сингулярно возмущённой параболической задачи с краевыми условиями Дирихле построено и обосновано асимптотическое разложение периодического по времени решения с пограничными слоями вблизи концов отрезка в случае, когда вырожденное уравнение имеет двукратный корень. Поведение решения в пограничных слоях и сам алгоритм построения асимптотики существенно отличаются от случая однократного корня вырожденного уравнения. Исследован также вопрос об устойчивости периодического решения и области его притяжения

Asymptotics, Stability, and Region of Attraction of Periodic Solution to a Singularly Perturbed Parabolic Problem with Double Root of a Degenerate Equation

For a singularly perturbed parabolic problem with Dirichlet boundary conditions, the asymptotic decomposition of a solution periodic in time and with boundary layers near the ends of the segment where the degenerate equation has a double root is constructed and substantiated. The construction algorithm for the asymptotics and the behavior of the solution in the boundary layers turn out to differ significantly as compared to the case of a simple root of a degenerate equation. The stability of the periodic solution and its region of attraction are also studied

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Applications of the blow-up technique in singularly perturbed chemical kinetics

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

Geometric Singular Perturbation Theory and Averaging: Analysing Torus Canards in Neural Models

Neuronal bursting, an oscillatory pattern of repeated spikes interspersed with periods of rest, is a pervasive phenomenon in brain function which is used to relay information in the body. Mathematical models of bursting typically consist of singularly perturbed systems of ordinary differential equations, which are well suited to analysis by geometric singular perturbation theory (GSPT). There are numerous types of bursting models, which are classified by a slow/fast decomposition and identification of fast subsystem bifurcation structures. Of interest are so-called fold/fold-cycle bursters, where burst initiation (termination) occurs at a fold of equilibria (periodic orbits), respectively. Such bursting models permit torus canards, special solutions which track a repelling fast subsystem manifold of periodic orbits. In this thesis we analyse the Wilson-Cowan-Izhikevich (WCI) and Butera models, two fold/fold-cycle bursters. Using numerical averaging and GSPT, we construct an averaged slow subsystem and identify the bifurcations corresponding to the transitions between bursting and spiking activity patterns. In both models we find that the transition involves toral folded singularities (TFS), averaged counterparts of folded singularities. In the WCI model, we show that the transition occurs at a degenerate TFS, resulting in a torus canard explosion, reminiscent of a classic canard explosion in the van der Pol oscillator. The TFS identified in the Butera model are generic, and using numerical continuation methods, we continue them and construct averaged bifurcation diagrams. We find three types of folded-saddle node (FSN) bifurcations which mediate transitions between activity patterns: FSN type I, II, and III. Type III is novel and studied here for the first time. We utilise the blow-up technique and dynamic bifurcation theory to extend current canard theory to the FSN III

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included