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Heteroclinic

In a moving co-ordinate system, the traveling wave equations typically reduce to a system of parameterized nonlinear ordinary differential equations. The solutions of this system corresponding to pulses and fronts for the original reaction-diffusion equation are called homoclinic and heteroclinic orbits, correspondingly, or just connecting orbits. [Pg.675]

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). [Pg.507]

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). [Pg.507]

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. [Pg.337]

Suppose that the unstable manifold of a NHIM intersects with the stable manifold of another NHIM (or the same NHIM) such intersections are called heteroclinic (or homoclinic). This means that there exists a path that connects these two NHIMs (or a path that leaves from and comes back to the NHIM). Thus, their intersections offer the information on how the NHIMs are connected. [Pg.339]

Here, we limit our argument to a system with a homoclinic connection—that is, a separatrix connecting a saddle with itself. The following argument can be straightforwardly extended to a system with a heteroclinic connection— that is, a separatrix connecting different saddles. [Pg.361]

As shown in Appendix B, item 1 is proven by using the Melnikov method. It means that the heteroclinic-like entanglement between the complexihed stable manifold and the initial time plane t occurs. [Pg.418]

We begin the treatment of Theorem 5.1(iv) by showing that the part of the one-dimensional unstable manifold of contained in 2 is a heteroclinic orbit connecting to Ei-... [Pg.200]

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. ... [Pg.166]

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. [Pg.166]

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... [Pg.171]

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. [Pg.508]

Homoclinic (or heteroclinic) tangency would be a common phenomenon for systems of more than two degrees of freedom. Therefore, its role in reaction dynamics has to be taken seriously. Toda [4] noticed that the homoclinic tangency would lead to crisis where a transition between chaos... [Pg.156]

One such property is the heteroclinic (or homoclinic) intersection between stable and unstable manifolds. [Pg.169]

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. [Pg.170]

This reasoning is not limited to homoclinic intersections. This argument can be easily extended to include heteroclinic intersections where stable and unstable manifolds may have different dimensions. [Pg.172]

The above reasoning is not limited to homoclinic tangency, but also can be applied to heteroclinic tangency such as that shown in Figure 3.12(b). [Pg.173]

Figure 3.12. (a) An example of homoclinic tangency. (b) An example of heteroclinic tangency. Here, H, Hj, and H2 indicate saddle points. [Pg.174]

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. [Pg.176]

In Figure 3.16, the time evolution of a wave packet is shown for y 0.3jt [17]. Here, the wave packet splits into two parts one directly approaches the dissociation limit and the other starts to take the path to internal rotation. These two paths correspond to the homoclinic intersection of the dissociation limit and the heteroclinic intersection, respectively. The superposition of these paths results in the split of wave packets in quantum mechanics. [Pg.180]

When there exist multiple paths with different decay times, the dissociation processes would exhibit multiexponential decay. In Figure 3.17, the time dependences of the remaining probabilities of wave packets are shown for different values of y [17], This finding indicates that the dissociation consists of two processes with different decay times. The faster one corresponds to the homoclinic intersection, and the slower one to the heteroclinic intersection. [Pg.180]


See other pages where Heteroclinic is mentioned: [Pg.287]    [Pg.123]    [Pg.329]    [Pg.193]    [Pg.247]    [Pg.250]    [Pg.306]    [Pg.310]    [Pg.405]    [Pg.406]    [Pg.427]    [Pg.427]    [Pg.484]    [Pg.45]    [Pg.475]    [Pg.582]    [Pg.170]    [Pg.171]    [Pg.190]    [Pg.190]    [Pg.122]    [Pg.138]    [Pg.156]    [Pg.179]   
See also in sourсe #XX -- [ Pg.397 , Pg.415 , Pg.424 , Pg.496 , Pg.520 ]




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A heteroclinic trajectory

Heteroclinic connection

Heteroclinic cycle

Heteroclinic fixed point

Heteroclinic orbit

Heteroclinic point

Heteroclinic trajectory

Transverse heteroclinic point

Unstable heteroclinic cycle

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