On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system

In this paper, infinite homoclinic orbits existing in the Lorenz system are analytically presented. Such homoclinic orbits are induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. A period-doubling bifurcation tree of period-1, period-2, and period-4 motions are generated as an example. The homoclinic orbits relative to unstable period-1, period-2, and period-4 motions are determined. Illustrations of homoclinic orbits and periodic orbits are given. This study presents how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.