On the Unresolved Cases of Convergence of Bidimensional Slow Discrete Dynamical Systems

Abstract

In this work we complete the analysis of convergence for nonhyperbolic fixed points of two dimensional discrete dynamical systems. We analyze all different scenarios of systems with isolated equilibria whose linearized part has eigenvalues of norm one, except for those with real eigenvalues equal to one, which have been studied previously. We also include the non-diagonalizable case and the case of mixing eigenvalues, where only one real eigenvalue has absolute value equal to one. Different techniques have been applied depending on the hyperbolic scenario. Finally, many two dimensional examples are presented in order to illustrate the diverse orbit behaviors and the number of iterations required to converge.