Heteroclinic cycle

Surprisingly, even non-rough systems of codimension one may have infinitely many moduli. Of course, since the models of nonlinear dynamics are explicitly defined dynamical systems with a finite set of parameters, this creates a new obstacle which the classical bifurcation theory has not nm into. Although the case of homoclinic loops of codimension one does not introduce any principal problem, nevertheless codimensions two and higher are much less trivial as, for example, in the case of a homoclinic or heteroclinic cycle including a saddle-focus where the structure of the bifurcation diagrams is directly determined by the specific values of the corresponding moduli. 

The violation of structural stability in Morse-Smale systems is caused by the bifurcations of equilibrium states, or periodic orbits, by the appearance of homoclinic trajectories and heteroclinic cycles, and by the breakdown of transversality condition for heteroclinic connections. However, we remark that some of these situations may lead us out from the Morse-Smale class moreover, some of them, under rather simple assumptions, may inevitably cause complex dynamics, thereby indicating that the system is already away from the set of Morse-Smale systems. 

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

Note that in many special cases attention is restricted to the study of the smaller spaces of systems, e.g. systems with some specified symmetries, Hamiltonian systems, etc. In view of that, the notion of structural stability in, say, Hamiltonian systems with one-degree-of-freedom becomes completely meaningful. So, for example, equilibrium states such as centers and saddles of such systems, become structurally stable. Moreover, if there are no heteroclinic cycles containing different saddles, we can naturally distinguish such systems as rough in the set of all systems of the given class. 

Fig. 10.6.2. The fixed point is a center at e = 0. The saddle fixed points form a heteroclinic cycle.

We study some homoclinic bifurcations of codimension two in Secs. 13.6. In Sec. 13.7, we review the results on the bifurcations of a homoclinic-8, and on the simplest heteroclinic cycles. 

In this and the following sections we will review some codimension-two bifurcations of homoclinic loops and heteroclinic cycles which occur in various models. 

In this section, we will review the bifurcations of a homoclinic-8, as well as of heteroclinic cycles including a pair of saddles such that they do not induce complex dynamics. We skip all proofs here just because our goal is only to... 

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... 

We assume throughout this section that the saddle values are negative in both saddles. In this case, no more than one periodic orbit can bifurcate from the heteroclinic cycle. Moreover, this unique orbit is stable (attracting). 

Fig. 13.7.16. Three types of heteroclinic cycles (a) Ai 2 > 0 (b) A 2 < 0 (c) A semiori-entable heteroclinic connection between two saddles on a Mobius band A < 0, A2 > 0.

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. 

Another case studied in [121] corresponds to the bifurcations of a heteroclinic cycle when the saddle values have opposite signs at equilibrium state Oi and O2 (the case where both saddle values are positive leads either to complex dynamics, if 0 and O2 are both saddle-foci, or reduces to the preceding one by a reversal of time and reduction to the invariant manifold). The main assumption here is that both 0 and O2 are simple saddles (not saddle-foci). 

Let i/i and 1/2 be the saddle indices at Oi and O2, respectively. Assume that 1, 1 2 1 and uii/2 1. Then, no more than two periodic orbits can bifurcate from the heteroclinic cycle under consideration. 

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. 

Fig. 13.7.24. A heteroclinic cycle between two saiddles. Notice that the heteroclinic trajectory Fi connecting 0 with O2 is structurally stable.

---