Homoclinic orbits and chaos in the generalized Lorenz system (Q2284925)

The paper investigates the existence of homoclinic orbits of the generalized Lorenz system \(\dot{x}=-s_{1}x+s_{2}y,\) \(\dot{y}=Rx+dy-xz,\) \(\dot{z}=-qz+xy,\) where \(\dot{}=\frac{d}{dt},s_{1}>d>0,q>0\) and the divergence \(\nabla =-s_{1}+d-q<0.\) The stability of the nonhyperbolic equilibrium is analyzed by center manifold theory. When the origin is a saddle, the nonexistence of homoclinic orbits are obtained by constructing Lyapunov functions. The existence of homoclinic orbits is investigated by the Fishing Principle. As a result, sufficient and necessary conditions for the existence of homoclinic orbits associated with the origin of the generalized Lorenz system are given. In case when the homoclinic orbits break, such system shows a chaotic attractor and the chaos is in the sense of generalized Shil'nikov homoclinic criteria.