Heteroclinic trajectory

The trajectory q pl (t) is determined by minimizing S in (20) on the set of all classical deterministic trajectories determined by the Hamiltonian H (37), that start on a stable limit cycle as t — —oo and terminate on a saddle cycle as t > oo. That is, qopt(t) is a heteroclinic trajectory of the system (37) with minimum action, where the minimum is understood in the sense indicated, and the escape probability assumes the form P exp( S/D). We note that the existence of optimal escape trajectories and the validity of the Hamiltonian formalism have been confirmed experimentally for a number of nonchaotic systems (see Refs. 62, 95, 112, 132, and 172 and references cited therein). 

If the noise is weak, then the probability P exp —S/D) to escape along the optimal trajectory is exponentially small, but it is exponentially greater than the escape probability along any other trajectory, including along other heteroclinic trajectories of the system (37). 

Notice that the twin saddle points are joined by a pair of trajectories. They are called heterocUnic trajectories or saddle connections. Like homoclinic orbits, heteroclinic trajectories are much more common in reversible or conservative systems than in other types of systems. ... 

Although we have relied on the computer to plot Figure 6.6.4, it can be sketched on the basis of qualitative reasoning alone. For example, the existence of the heteroclinic trajectories can be deduced rigorously using reversibility arguments (Exercise 6.6.6). The next example illustrates the spirit of such arguments. 

There are several advantages to the cylindrical representation. Now the periodic whirling motions look periodic—they are the closed orbits that encircle the cylinder for E>1. Also, it becomes obvious that the saddle points in Figure 6.7.3 are all the same physical state (an inverted pendulum at rest). The heteroclinic trajectories of Figure... 

It is very convenient to represent the eigen solutions of (23) as trajectories in the phase space. Then solitary and level jump waves are represented by homoclinic and heteroclinic trajectories accordingly. Periodic and biperiodic solutions in phase space by circles and invariant torii are represented. 

The last condition may be reformulated as the absence of homoclinic and heteroclinic trajectories. 

Among all heteroclinic trajectories one may select some special ones which play a central role. 

Definition 7.12. A heteroclinic trajectory T is said to be special if there exists a neighborhood U of its closure T which contains no other heteroclinic trajectories but T. 

It is obvious that all heteroclinic trajectories of three-dimensional Morse-Smale flows are that special. This is also true for two-dimensional diffeomor-phisms. 

The principal feature of Morse-Smale systems which distinguishes them from Andronov-Pontryagin systems is that the former may have infinitely many special heteroclinic trajectories. As an example, let us consider a two-dimensional diffeomorphism with three fixed points of the saddle type denoted by Oi, O and O2- Suppose that O Wq 0 and n Wq 0, the... 

Let us now apply the A-lemma (see Sec. 3.7). Choose a small neighborhood U of the point M. It follows that the intersection U Pi Wq consists of a countable set of curves Ik (A = 1,..., 00) accumulating smoothly to Wq, as shown in Fig. 7.6.2. As Wq and Wq intersect each other transversely, then Wq intersects each Ik at the points Mk starting from some number ko. The points Mfc are heteroclinic too and correspond to different heteroclinic trajectories which have Oi and O2 as an Q-limit and an cj-limit points, respectively. 

Theorem 7.13. (Afraimovich and Shilnikov [2]) The intersection Wq C Wq possesses infinitely many heteroclinic trajectories if, and only if the closure Wgi Pi Wq contains a periodic orbit L of type m- -l,n — m), other than Qi and Q2>... 

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. 

Another example is a family of two-dimensional C -smooth diffeomor-phisms whose non-wandering set does not change until the boundary of Morse-Smale diffeomorphisms is reached. The situation is illustrated in Fig. 8.2.3. The two fixed points 0 and O2 have positive multipliers, and Wq contacts Wq along a heteroclinic trajectory, and so do Wq and. This example... 

Fig. 8.3.1. A nontransverse heteroclinic trajectory between two saddle fixed points.

The value 9 is also a modulus of topological equivalence in the case of a three-dimensional fiow which has two saddle periodic orbits such that an unstable manifold of one periodic orbit has a quadratic tangency with a stable manifold of another orbit along a heteroclinic trajectory. 

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.

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

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.

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