Bifurcation Trees of (1:2)- Asymmetric Periodic Motions with Corresponding Infinite Homoclinic Orbits in the Lorenz System

Abstract

In this paper, bifurcation dynamics and infinite homoclinic orbits of (1:2)- asymmetric periodic motions in the Lorenz systems are studied through the discrete mapping method. The infinite homoclinic orbits pertaining to the unstable periodic motions on the bifurcation trees of (1:2)-asymmetric periodic motions to chaos are determined. The stability and bifurcations of periodic motions are determined through the eigenvalue analysis. A bifurcation tree of (1:2) asymmetric periodic motions, varying with the Rayleigh number, is presented by discrete nodes, and the corresponding harmonic frequency-amplitude characteristics of periodic motions on the bifurcation tree are discussed through finite Fourier series analysis. A sequence of period-doubling bifurcations is observed on the bifurcation tree of the (1:2)- asymmetric periodic motion to chaos. The scenario of (1:2)- asymmetric period-1, period-2, period-4 motions to chaos scenario is illustrated. Homoclinic orbits are as appearance or vanishing of unstable periodic motions on the bifurcation tree. Illustrations of periodic motions and homoclinic orbits are completed for intuitive demonstrations of the corresponding topological structures. This study extends the initial study of the bifurcation tree from the (1:1)-symmetric periodic motions to asymmetric periodic motions then to chaos in the Lorenz system, and the corresponding infinite homoclinic orbits induced by all unstable periodic motions can be determined. Such results can enrich the understanding of bifurcation dynamics of asymmetric periodic motions to chaos in the Lorenz system.