Heteroclinic point

Figure 16 Sketches of numerically generated separatrices on the Poincare map. After extracting the manodromy matrix of a hyperbolic fixed point (p2> the asymptotic eigenvectors W+ and W can be obtained. (A) If no other fix points are nearby in the chaotic sea, the separatrix branch formed by repeated mappings in positive time of points initially on will eventually meet the branch formed by repeated mappings in negative time of points initially on W at a single point hj (called a homoclinic point). The closed curve so generated is the separatrix 5. (B) If a second fixed point (p2> 2)2 associated with a different periodic orbit is nearby, a separatrix 5 may be formed by the intersection of branches arising from the two orbits at two points hi and (called heteroclinic points).

The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped a = 0 and unperturbed 7 = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter 7 = 0 and the dissipation parameter a = 0 in (3.45), and are... 

The definition of the homoclinic and heteroclinic points needs first the introduction of hyperbolic and elliptic points. A two-dimensional flow always consists of hyperbolic and/or elliptic points (Fig. 6.26). At the hyperbolic point the fluid moves toward it in one direction and away from it in another direction. At an elliptic point the fluid moves in closed pathlines. A periodic point is defined as the point at which a particle in a periodic flow returns after a number of periods. The number of periods defines also the order of the periodic point, as periodic point of period 1, 2, and so on. Note that the periodic elliptic points should be avoided should we want enhanced mixing. A point where the outflow of one hyperbolic point intersects the inflow of another hyperbolic flow is called transverse heteroclinic point. When the inflow and outflow refer to the same hyperbolic point, the point is called transverse homoclinic point. 

Time-dependent, periodic, two-dimensional flows can result in streamlines that in one flow pattern cross the streamlines in another pattern, and this may lead to the stretching-and-folding mechanism that we discussed earlier, which results in very efficient mixing. In such flow situations, the outflow associated with a hyperbolic point can cross the region of inflow of the same or another hyperbolic point, leading, respectively, to homoclinic or heteroclinic intersections these are the fingerprints of chaos. 

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. ... 

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... 

Quasiperiodicity is significant because it is a new type of long-term behavior. Unlike the earlier entries (fixed point, closed orbit, homoclinic and heteroclinic orbits and cycles), quasiperiodicity occurs only on the torus. 

In Figure 3.10(b), the point (qi = q°, Pi = 0) is a saddle point of the potential. While those orbits that asymptotically approach the saddle point constitute the stable manifold, those that asymptotically leave the saddle form the unstable manifold. These two manifolds do not in general coincide with each other. For systems of one degree of freedom under a periodic external force and those of two degrees of freedom, these two manifolds have intersections. When the two manifolds have the same saddle point in common as shown Figure 3.10(h), their intersections are called homoclinic. When they do not, their intersections are called heteroclinic. 

Figure 3.12. (a) An example of homoclinic tangency. (b) An example of heteroclinic tangency. Here, H, Hj, and H2 indicate saddle points. 

The first method comes from the idea that the connections among normally hyperbolic invariant manifolds would form a network, which means that one manifold would be connected with multiple manifolds through homoclinic or heteroclinic intersections. Then, a tangency would signify a location in the phase space where their connections change. This idea offers a clue to understand, based on dynamics, those reactions where one transition state is connected with multiple transition states. In these reaction processes, the branching points of the reaction paths and the reaction rates to each of them are important We expect that analysis of the network is the first step toward this direction. 

In Sections IV and V, we discussed these two processes from the viewpoint of chaos. In these discussions, the following points are of importance. As for the barrier crossing, intersection (either homoclinic or heteroclinic) between stable and unstable manifolds offers global information on the reaction paths. It not only includes how transition states are connected with each other, but also reveals how reaction paths bifurcate. Here, the concepts of normally hyperbolic invariant manifolds and crisis are essential. With regard to IVR, the concept of Arnold web is crucial. Then, we suggest that coarse-grained features of the Arnold webs should be studied. In particular, the hierarchy of the web and whether the web is uniformly dense or not would play an important role. 

V < 2V. The state (1,0) is always a saddle point. To be physically acceptable, a front must always be nonnegative. Consequently, only nonnegative heteroclinic orbits are acceptable. Such orbits can only exist if (0,0) is a stable node. In other words, fronts only exist for v > 2 /z5r. Since there exists a heteroclinic connection or front for each value of v with v > 2 /Dt, this analysis does not yield a unique propagating velocity. In fact, the front velocity v depends on the initial condition, specifically on the tail of the initial condition. 

The nonnegative heteroclinic orbit will persist as v decreases, as long as no bifurcation occurs in the vector field of (5.64a) and (5.64c) and (1/2, 1 /2) remains a saddle poinf and (0, 0) a sfable node, whose eigenvectors lie strictly within the positive quadrant. For w < y, the eigenvalues for the fixed point (0, 0) are... 

This result is due to Palis, who had fotmd that two-dimensional diffeomor-phisms with a heteroclinic orbit at whose points an unstable manifold of one saddle fixed point has a quadratic tangency with a stable manifold of another saddle fixed point can be topologically conjugated locally only if the values of some continuous invariants coincide. These continuous invariants are called moduli. Some other non-rough examples where moduli of topological conju-gacy arise are presented in Sec. 8.3. 

The closure of an unclosed Poisson-stable trajectory whose return times are unbounded for some e > 0 is called a quasiminimal set. A quasiminimal set contains, besides Poisson-stable trajectories which are dense everywhere in it, some other invariant and closed subsets. These may be equilibrium states, periodic orbits, non-resonant invariant tori, other minimal sets, homoclinic and heteroclinic orbits, etc., among which a P-trajectory is wandering. This gives a clue to why the recurrent times of the non-trivial unclosed P-trajectory are unbounded. Furthermore, this also points out that Poisson-stable trajectories of a quasiminimal set, due to their unpredictable behavior in time, are of... 

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. 

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