The existence of homoclinic orbits in the Lorenz system via the undetermined coefficient method

Abstract

In this paper, we are concerned about homoclinic orbits of the Lorenz system{x˙=−σx+σy,y˙=rx−y−xz,z˙=−bz+xy,where σ, b, r > 0 are three parameters. Firstly, by Fredholm alternative theorem and Taylor expansions, one can figure out the local invariant manifolds of the trivial equilibrium E0=(0,0,0), including the stable and unstable ones. Secondly, the undetermined coefficient method is used to find out the negative part of the homoclinic orbit, which is supposed to have the form of exponential type series. Such negative semiorbit together with the positive semiorbit that starts from the locally stable manifold consists of a homoclinic loop. As these two semiorbits should be continuous at the intersection point, and the corresponding series should be convergent, one can establish the conditions for the existence of homoclinic orbits. Finally, as an application, we will exhibit the existence theorem of homoclinic orbits for the Lorenz system with the parameters σ=10,b=8/3,r=13.926.