Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle–focus and Saddle Under Reversible Condition

Abstract

In this paper, we study the dynamical behavior for a 4–dimensional reversible system near its heteroclinic loop connecting a saddle–focus and a saddle. The existence of infinitely many reversible 1–homoclinic orbits to the saddle and 2–homoclinic orbits to the saddle–focus is shown. And it is also proved that, corresponding to each 1–homoclinic (resp. 2–homoclinic) orbit Γ, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1–periodic (resp. 2–periodic) and accumulate onto Γ. Moreover, each 2–homoclinic orbit may be also accumulated by a sequence of reversible 4–homoclinic orbits.