Takens–Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system

Abstract

In this work we study Takens–Bogdanov bifurcations of equilibria and periodic orbits in the classical Lorenz system, allowing the parameters to take any real value. First, by computing the corresponding normal form we determine where the Takens–Bogdanov bifurcation of equilibria is non-degenerate, namely of homoclinic or of heteroclinic type. The transition between these two types occurs by means of a triple-zero singularity. Moreover, we demonstrate that a degenerate homoclinic-type Takens–Bogdanov bifurcation of infinite codimension occurs. Secondly, taking advantage of the above analytical results, we carry out a numerical study of the Lorenz system. In this way, we find several kinds of degenerate homoclinic and heteroclinic connections as well as Takens–Bogdanov bifurcations of periodic orbits. The existence of these codimension-two degeneracies, that organize the symmetry-breaking, period-doubling, saddle-node and torus bifurcations undergone by the corresponding periodic orbits, guarantees in some cases the presence of Shilnikov chaos. We also show the existence of a codimension-three homoclinic connection that together with the triple-zero degeneracy act as main organizing centers in the parameter space of the Lorenz system. Finally, we obtain interesting information on the bifurcation sets of the widely studied Chen and Lü systems, taking into account that they are, generically, particular cases of the Lorenz system, as can be proved with a linear scaling in time and state variables. We remark that the heteroclinic case of the Takens–Bogdanov bifurcation in the Lorenz system was found in the literature, in a region with negative parameters, in the study of a thermosolutal convection model and in the analysis of traveling-wave solutions of the Maxwell–Bloch equations.