Teoría cualitativa de los sistemas dinámicos no lineales y el modelo de FitzHugh-Nagumo

Abstract

The present work is divided into two parts. First, it examines the fundamental concepts of qualitative theory for analyzing nonlinear dynamic systems, including the theorems of Hartman-Grobman, Lyapunov, and the Bendixon's criterion. Subsequently, these theoretical elements are applied to the analysis of the Fitzhugh-Nagumo model. Special attention is given to the local analysis of the system, represented by the following set of equations: x' = y + x - x³/3 + I and y' = -x + a - by where x' and y' denote the temporal derivatives of x and y respectively, a, b, and I are real parameters (Murray, 2002). A variety of different behaviors are identified, ranging from hyperbolic and non-hyperbolic points to spirals, nodes, saddles, and cycles.