Bifurcation Of Rough Heteroclinc Loop With Orbit And Inclination Flips

In this paper, we are concerned with the heteroclinic loop (Γ = Γ₁ ∪Γ₂) bifurcations accompanied by orbit and inclination flips under the roughness condition, where Γ₁ is a heteroclinic orbit with orbit flip (that is to say, Γ₁ enters the hyperbolic critical point as t→+∞ along the tangential direction of the strong stable manifold); and Γ₂ is of inclination flip(that is to say, the strong stable manifold of a saddle is asymptotic to the tangential direction of another saddle's weak stable manifold along the inverse directon of Γ₂). Owing to the complexity of the higher codimension bifurcation, we restrict our results under the generic and untrivial roughness condition.We obtain Poincare map and bifurcation equations (2.2.7) by establishing the local coordinate and using the transformation of Silnikov coordinate. Moreover, the existence of periodic orbits or homoclinic orbits of (2.1.1) is equal to the existence of positive solutions (s₁,s₂) of the bifurcation equations (s₁,s₂ ≥ 0). The following results can be derived: The existence of one 1-periodic orbit and one 1-homoclinic orbit is obtained if the heteroclinic loop is twisted, where periodic orbit and homoclinic orbit don't exist in the same time; but if the heteroclinic loop is nontwisted, the existence of one 1-periodic orbit, one 1-homoclinic orbit, two 1-periodic orbits, one double 1-periodic orbits and the co-existence of periodic orbit and homoclinic orbit in particular situations is obtained, respectively, approximate expressions of the corresponding bifurcation curves (or surfaces) are also given. The corresponding four bifurcation graphs can be given with clarity by summarizing all the above results (see Figure 5-8).Finally, non-existence of 2-homoclinic orbit bifurcation curves (or surfaces) is obtained by deeply studying the bifurcation equations in case Γ is absolutely nontwisted.