Homoclinic-8 connection

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. 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

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. 

The system (7) with e = 0 is referred as unperturbed system. About it we shall assume that it possesses a hyperbolic fixed point xQyh connected to itself by a homoclinic orbit Xh(t) = x (t), x (t)). 

This orbit is connected to itself by a pair of 2-dimensional homoclinic manifolds given by... 

The double-zero eigenvalue points, such as M, represent the coalescence of Hopf bifurcation and stationary-state turning points. As mentioned above, they thus represent the points at which the Hopf bifurcation loci begin and end. They also have other significance. Such points correspond to the beginning or end of loci of homoclinic orbits. For the present model, with the given choices of k1 and k2, there are two curves of homoclinic orbit points, one connecting M to N, the other K to L, as shown schematically in Fig. 12.7. 

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. 

An important event in the phase plane occurs if the inset to a saddle manages to join up with an outset from the same saddle point. This then gives rise to a closed loop with the saddle point lying on it as a corner . Such a loop is known as a homoclinic orbit as it forms a path connecting... 

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. ... 

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. 

A special solution of Eqs. (7),(8),(9) which connects two different steady states (a heteroclinic solution) then represents a front wave and a solution doubly asymptotic to a single steady state (a homoclinic orbit) represents a pulse wave. 

Fig. 7.35 Homoclinic trajectraies connecting the (3, 1) saddle-nodes on the separatrix of the torus about the Li atom in the LiH molecule. The arrows indicate the direction of the eigenvectos of the transposed Jacobian matrix V/ at the stagnation points. An asymptotic wavy line flows across the stagnation loop, about its centre...

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

Similarly, if there were a separatrix loop to a saddle at // = 0, it would be split for some non-zero /i, as shown in Fig. 7.1.2. We see that an arbitrarily small smooth perturbation of the vector field will modify the phase portrait of a system with a homoclinic loop or a heteroclinic connection this obviously means that such a system is non-rough. 

Fig. 8.1.6. (a) A structurally unstable saddle connection after the disappearance of a saddle-node cycle in Pig. 8.1.4 (b) Phase plane after the splitting of the homoclinic loop in Fig. 8.1.5. 

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. 8.4.1. (a) A heteroclinic connection between a saddle O2 and saddle-focus Oi (b) a heteroclinic connection between two saddle-foci Oi,2 (c) a homoclinic figure-eight to a saddle-focxis. 

The bifurcation diagram for the case where both Oi and O2 are saddle-foci is shown in Fig. 13.7.17. Here, the curves L and L2 which correspond to the homoclinic loops intersect the curves C21 / i = 0 and C 2 a 2 = 0 infinitely many times. Next, for each A = 0,1,2,..., for any two neighboring points of intersection of L with a connected component of L2 with a connected component of C12) such that the inequality hi fjL2) > hki2 fJ>2) (respectively, /i2(mi) > hk2i f )) holds between these points, there is a component of the curve (respectively C i ) which connects these points. In turn,... 

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