Superluminal periodic orbits in the Lorenz system

Abstract

In this work we present, for the Lorenz system, analytical and numerical results on the existence of periodic orbits with unbounded amplitude and whose period tends to zero. Since a particle moving on these periodic orbits would be faster-than-light, we call them superluminal periodic orbits. To achieve this goal, we first find analytical expressions for the period in three different situations, where Hopf and Takens–Bogdanov bifurcations of infinite codimension occur. Thus, taking limit in the corresponding expressions allows to demonstrate the existence of superluminal periodic orbits for finite values of the parameter ρ (in a region where the other two parameters σ and b are negative). Moreover, we numerically show, in other two different cases of physical interest, that these orbits also exist when the parameter ρ tends to infinity. Finally, the presence of superluminal periodic orbits in the widely studied Chen and Lü systems follows directly from our results, 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.