Dynamics Related to the Presence of Canard Solutions

New Mexico State University, Las Cruces NM

Investigators

Abstract

The context of this project is singularly perturbed differential equations. Sample equations to be considered include coupled oscillators and models of oscillatory chemical reactions. A characteristic feature of singularly perturbed equations is the presence of slow/fast dynamics. Slow/fast systems often possess locally invariant slow manifolds, where trajectories evolve on the slow time scale. As trajectories leave locally invariant slow manifolds they typically follow fast directions. Canard solutions are exceptional trajectories that pass from attracting to repelling slow manifolds. It is known that these solutions separate invariant or locally invariant regions of different dynamic behavior. For his reason canard solutions provide the key to understanding certain types of complex oscillatory and/or chaotic dynamics. The goal of the project is to understand the dynamics related to the presence of canard solutions. The methods that will be used in the analysis are based on geometric theory of dynamical systems augmented by a desingularization procedure from algebraic geometry known as the blow-up method. The need for desingularization is explained by the fact that canards pass through regions where the slow dynamics is not hyperbolic. An efficient approach is to break up the analysis into local, near the points of non-hyperbolicity, and global, where standard methods (Fenichel theory) apply. Slow/fast dynamics are observed in the context of a number of natural phenomena. For example, in the context of the firing of a neuron, the membrane potential is a fast varying variable, while some of the other variables, related to the concentration of some of the ions, like potassium, vary slowly. For a simple oscillator (cell) consisting of one fast variable and one slow variable there are two basic types of oscillations: small oscillations, which correspond to uniformly slow motions small in amplitude, and relaxation oscillations, which are a combination of a slow evolution and fast jumps. Systems with many interacting separate components are often modeled as coupled cells, with each component modeled by a simple cell (oscillator) and the interaction between cells modeled by appropriate coupling terms. Such models arise in a large variety of applications, including biotechnology, chemical engineering, and neurobiology. One of the fundamental issues in the analysis of such systems is to understand the patterns of oscillation. Interesting examples of such patterns are mixed mode oscillations, which correspond to switching between small oscillations and relaxation oscillations, localized solutions, which occur when a cluster of cells performs relaxation oscillations while the remaining cells perform small oscillations, and multi-periodic solutions, for which the frequency of the oscillation of some of the cells is a fraction of the frequency of the oscillation of the other cells. The goal of this project is to obtain a better understanding of the patterns of oscillation and possibly of more complicated dynamics by using the structure provided by the existence of exceptional solutions, called canard solutions, which occur on the boundaries of regions corresponding to different dynamic behaviors.