Non-degeneracy assumptions

The situation which we consider here is a particular case of Theorem 13.9 of the next section. It follows from this theorem (applied to the system in the reversed time) that a single saddle periodic orbit L is born from a homoclinic loop it has an m-dimensional stable manifold and a two-dimensional unstable manifold. This result is similar to Theorem 13.6. Note, however, that in the case of a negative saddle value the main result (the birth of a unique stable limit cycle) holds without any additional non-degeneracy requirements (the leading stable eigenvalue Ai is nowhere required to be simple or real). On the contrary, when the saddle value is positive, a violation of the non-degeneracy assumptions (1) and (2) leads to more bifurcations. We will study this problem in Sec. 13.6. 

Fig. 13.4.11. Behavior of the stable manifold Wf in backward time near a homoclinic loop to the saddle with real Ai, provided the non-degeneracy assumptions hold.

Let us now make two more non-degeneracy assumptions. Namely, we assume that... 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

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