Estudo da bifurcação de Neimark-Sacker

Abstract

This dissertation deals with a local bifurcation for planar smooth mappings, depending on a real parameter, called Neimark-Sacker bifurcation of codimension 1, which, in a certain sense, shares many similarities with the Hopf bifurcation for ordinary di erential equations. In both bifurcations, the change in stability of a xed point or equilibrium point, together with a transversality condition associated with certain eigenvalues of the Jacobian matrix evaluated at the point, along with one or more nondegeneracy conditions, allows the appearance or disappearance of an invariant closed curve by the dynamics in the phase portrait when the parameter is varied. This topic was chosen due to its importance in the study of discrete dynamical systems and applications in many scienti c areas. In this sense, the Theorem of Neimark-Sacker Bifurcation of codimension 1 is stated and proved in the planar case, and applied to the study of two well-known biological models, namely, the delayed logistic equation and the discrete predator-prey equation.