The Extension of(S)i'lnikov Homoclinic Theorem and Its Applications

摘要

The traditional Si'lnikov theorems provide analytic criteria for proving the existence of chaos in highdimensional autonomous systems. In this paper, the extension questions of the Si'lnikov homoclinic theorem and its applications are discussed. We establish two extended versions of the Si'lnikov homoclinic theorem and give a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In addition, we construct new three-dimensional chaotic systems which meet all the conditions in the extended Si'lnikov homoclinic theorem, and demonstrate the corresponding chaotic attractors numerically. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Si'lnikov theorems.