Some Routes to Chaos from Limit Cycle in the Forced Lorenz System

Abstract

Previously we have investigated the response of the Lorenz system under periodic perturba­ tion. In this paper, we focus on the transition from a limit cycle to chaos in the same system. The following three cases of the transition are studied: (I) Chaos through cascades of subharmonic bifurcation. In the periodic region, the stroboscopic mapping in the system is approximated by the Henon mapping. We obtain the Feigenbaum constant a = 4. 715. (II) Chaos due to the production of a homoclinic intersection in the stroboscopic picture. The system shows Intermittency. (III) Chaos due to the production of a pair of heteroclinic intersections in the stroboscopic picture. The system shows Intermittency. The resultant strange attract or has a positive two­ dimensional Lyapunov characteristic number. Power spectra and Lyapunov characteristic numbers are obtained numerically in each case. Furthermore, in connection with Case (II), we show that in a system which has a certain type of spatial symmetry such as is found in the Lorenz model, a symmetric solution cannot undergo subharmonic bifurcation.