Unstable heteroclinic cycle

Fig. 8.2.3. A structurally unstable heteroclinic cycle including two saddle fixed points.

Fig. 8.3.2. A structurally unstable heteroclinic trajectory connecting a saddle-focus and a saddle periodic orbit with positive multipliers, i.e. both manifolds of the saddle cycle are homeomorphic to a cylinder.

Let us now consider the case of a heteroclinic cycle with two saddles 0 and O2, Let the unstable manifolds of both saddles be one-dimensional and let an unstable separatrix Fi of Oi tend to O2 as t —> H-oo and an unstable... 

Note that all of these results (except for the subtle structure of the set of curves C12 in the case where Oi is a saddle-focus and O2 is a saddle) are proven for C -smooth systems. Therefore, just like in the case of a homoclinic-8, these results can be directly extended to the case where the unstable manifolds of Oi and O2 are multi-dimensional (but they must have equal dimensions in this case), provided that the conditions of Theorem 6.4 in Part I of this book, which guarantee the existence of an invariant -manifold near the heteroclinic cycle, are satisfied. 

Fig. 13.7.20. The bifurcation diagram for the case where both equilibrium states of the heteroclinic cycle are saddles (see Fig. 13.7.12) provided Ai,2 > 0, i/i > 1, 1/2 < 1 and U1U2 > 1- The system has one limit cycle in regions 1-3, two limit cycles in region 4, none in regions 5-7. A pair of limit cycles are born from a saddle-node on the curve SN the unstable one becomes a homoclinic loop on the curve L2, whereas the stable limit cycle terminates on Li.

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. 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

Theorem 13.9. (Shilnikov [134]) Let a saddle O with saddle value cr < 0 have a homoclinic loop F which satisfies the non-degeneracy conditions (1) and (2). Let U he a small neighborhood ofT. If the homoclinic loop splits inward on the invariant manifold Ad, then a single periodic orbit L with an n-dimensional unstable manifold will be bom. Furthermore the only orbits which stay in U for all times are the saddle O, the cycle L and a single heteroclinic orhi which is ot-limit to O and u)-limit to L. 

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