Resonances of periodic orbits in the Lorenz system

Abstract

Usually, the physical interest of the Lorenz system is restricted to the region where its three parameters are positive. However, this famous system appears, when $$\sigma <0$$ , in the study of a thermosolutal convection model and in the analysis of traveling-wave solutions of the Maxwell–Bloch equations. In this context, a Takens–Bogdanov bifurcation of heteroclinic type becomes an important organizing center. It has been very recently shown that the periodic orbit born in the Hopf bifurcation of the origin undergoes a torus bifurcation. In this paper we perform a detailed numerical study of the resonances of periodic orbits in the three-parameter Lorenz system, $$ \dot{x} = \sigma (y-x), \ \dot{y} = \rho x - y - xz, \ \dot{z} = -bz + xy, $$ when $$\sigma <0$$ and $$\rho , b >0$$ . The combination of numerical continuation methods and Poincare sections of the flow provides important information of how the resonances appear and evolve giving rise to a very rich dynamical and bifurcation scenario.