On šilnikov′s homoclinic-saddle-focus theorem

Consideration is given to the chaotic dynamics near an orbit homoclinic to a saddle-focus fixed point of šilnikov type. A new type of symbolic system is used to describe the dynamics on the unstable manifold and its relation to the šilnikov dynamics. Techniques based on the šilnikov method are developed for this unifying treatment. These new ideas and techniques should be applicable for a wide variety of homoclinically and heteroclinically related problems.