The mechanism of rotating waves in a ring of unidirectionally coupled Lorenz systems

Abstract

In this paper, we investigate the mechanism of rotating waves in a ring of unidirectionally coupled Lorenz systems. Rotating waves in our Lorenz systems are special cases of rotating-periodic solutions in nonlinear systems. Rotating-periodic solutions as a generalization of periodic solutions, have the form x(t+T)=Qx(t) with some orthogonal matrix Q. We establish the Hopf bifurcation theorem of rotating-periodic solutions in general odd-dimensional dynamical systems or systems coupled by multiple odd-dimensional subsystems. And by this bifurcation theorem, the existence of the rotating wave bifurcation is proved in the coupled Lorenz systems. In particular, the rotating waves can be periodic or quasi-periodic in our result.