Bifurcaciones locales y globales en sistemas dinámicos y autónomos

Abstract

In this thesis we analyze some aspects of the dynamics in several three-dimensional systems. In the first two chapters we study the singularities in the Lorenz system. In chapter 1 we carried out a complete study of the Hopf bifurcation experienced by both the equilibrium at the origin and the nontrivial equilibria, as well as their degeneracies. We prove the existence of bifurcations of codimension one, two and three, as well as regions of the parameter space where a degenerate Hopf bifurcation of infinite codimension occurs. In chapter 2 we study the Takens-Bogdanov bifurcation of the equilibrium at the origin. We determine for which parameter values it is of homoclinic or heteroclinic type, as well as its degeneracies. Concretely, we find a degeneracy of infinite codimension when it is of homoclinic type. Our detailed numerical study allows us to find Takens-Bogdanov bifurcations of periodic orbits. These codimension two degenerations organize bifurcations (symmetry breaking, period doubling, saddle-node and torus) in the corresponding periodic orbits, so that the presence of Shilnikov chaos is guaranteed in some cases. In chapter 3 we study a double zero bifurcation in a triparametric system that unfolds the triple zero bifurcation of the Lorenz system. We demonstrate that it organizes around it several types of bifurcations (transcritical, pitchfork, Hopf and heteroclinic cycle) and we provide the approximate expressions of the corresponding curves. The numerical study allows us to detect several homoclinic and heteroclinic degenerate connections, heteroclinic T-point cycles and chaotic attractors. Furthermore, we obtain four curves of global bifurcations of codimension two which are related to the triple-zero bifurcation. In chapter 4 we consider a two-parameter quadratic three-dimensional system with only six terms and two nonlinearities. First, we study the Hopf bifurcation of its only equilibrium. Then, a careful numerical study of the homoclinic connections, in the region where this equilibrium is real saddle, allows us to detect a homoclinic flip bifurcation of Cin type. This is the first example in the literature of a three-dimensional system exhibiting this bifurcation, which leads to the presence of chaotic behaviour. Finally, in chapter 5, to complement the results obtained in chapter 3, we are interested in the analysis of the double-zero bifurcation that the equilibrium of the origin undergoes in the Lorenz system. Because in this case the equilibrium is not isolated, we cannot use the usual analytical techniques. To circumvent this difficulty, we add a new quadratic term in the third equation of the Lorenz system. The theoretical study of this singularity in this quasi-Lorenz system guarantees, for certain values of the parameters, the existence of a heteroclinic cycle. Next, the numerical analysis in the parameter space of this heteroclinic connection allows us to detect certain degeneracies of the global connections which act as organizing centers of the complex dynamics of the system. If we analyze the evolution of this set of degeneracies when the coefficient of the new term introduced approaches zero, we can understand, in the Lorenz model, the origin of an infinite succession of global connections that was already mentioned in the literature but whose origin was unknown.