A NOTE ON THE COEXISTENCE OF HOMOCLINIC ORBIT AND SADDLE FOCUS POINT FOR SOME TYPICAL CHAOTIC SYSTEMS

The coexistence of homoclinic orbit and saddle focus point is the basic assumption in Shil'nikov homoclinic theorem. We attempt to study the existence of homoclinic orbit to saddle focus point and give the necessary conditions for it. Firstly, the geometrical properties of homoclinic orbit to saddle focus point are exposed by some lemmas which are used to drive the main theorem. Consequently, the necessary conditions for the existence of homoclinic orbit to saddle focus point are obtained. The result is applied to Lorenz-type systems. Finally, the conclusions for some typical chaotic systems is presented.