Homoclinic manifold

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

On this interval, the homoclinic-8 bifurcates in the same way as in the Khorozov-Takens normal form. Both loops, which form the homoclinic-8 are orientable. The dimension of the center homoclinic manifold is equal to 2. The third dimension does not yet play a significant role. Therefore, it follows from the results in Sec. 13.7 that on the right of HS, there are two unstable cycles cycles 1 and 2 in Fig. 13.7.9). To the left of HS, a symmetric saddle periodic orbit (cycle 12) bifurcates from the homoclinic-8 (see also Fig. C.7.5). 

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... 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

FIGURE 9 Stroboscopic phase portraits for the points on figure 8 labelled (a)-(e). (a) Below the third root of unity point (labelled F in figure 8) the phase portrait is structurally a period three phase locked torus (b) above point F, the period I fixed point in the centre is now stable and the phase locked torus has disappeared (c)-(e) before, during, and after a period 3 homoclinic bifurcation to the right of point F oil cut, = 3.97 for each, and A/Ao = 5.90, 5.93 and 5.95 for (c)-(e) respectively. The period 3 phase locked torus is transformed to a free torus as the stable manifold of each saddle crosses the unstable manifold of an adjacent saddle. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

The bursting dynamics ends in a different type of process, referred to as a global (or homoclinic) bifurcation. In the interval of coexisting stable solutions, the stable manifold of (or the inset to) the saddle point defines the boundary of the basins of attraction for the stable node and limit cycle solutions. (The basin of attraction for a stable solution represents the set of initial conditions from which trajectories asymptotically approach the solution. The stable manifold to the saddle point is the set of points from which the trajectories go to the saddle point). When the limit cycle for increasing values of S hits its basin of attraction, it ceases to exist, and... 

Similar to unstable periodic orbits, an NHIM has stable and unstable manifolds that are of dimension 2 — 2 and are also structurally stable. Note that a union of the segments of the stable and unstable manifolds is also of dimension 2n — 2, which is only of dimension one less than the energy surface. Hence, as far as dimensionality is concerned, it is possible for a combination of the stable and unstable manifolds of an NHIM to divide the many-dimensional energy surface so that reaction flux can be dehned. However, unlike the fewdimensional case in which a union of the stable and unstable manifolds necessarily encloses a phase space region, a combination of the stable and unstable manifolds of an NHIM may not do so in a many-dimensional system. This phenomenon is called homoclinic tangency, and it is extensively discussed in a recent review article by Toda [17]. 

Also interesting is the dynamical behavior associated with the fixed point at infinity, that is, q,p) = (oo,0). Here we introduce the concept of homoclinic orbit, which is a trajectory that goes to an unstable fixed point in the past as well as in the future. A homoclinic orbit thus passes the intersection between the unstable and stable manifolds of a particular fixed point. Indeed, as shown in Fig. 6, these manifolds generate a so-called homoclinic web. In particular. Fig. 6a displays a Smale horseshoe giving a two-symbol subdynamics, indicating that the fixed point (oo,0) is not a saddle. Nevertheless, it is stUl unstable with distinct stable and unstable manifolds, with its dynamics much slower than that for a saddle. Figure 6b shows an example of a numerical plot of the stable and unstable manifolds. 

A beautiful classical theory of unimolecular isomerization called the reactive island theory (RIT) has been developed by DeLeon and Marston [23] and by DeLeon and co-workers [24,25]. In RIT the classical phase-space structures are analyzed in great detail. Indeed, the key observation in RIT is that different cylindrical manifolds in phase space can act as mediators of unimolecular conformational isomerization. Figure 23 illustrates homoclinic tangling of motion near an unstable periodic orbit in a system of two DOFs with a fixed point T, and it applies to a wide class of isomerization reaction with two stable isomer... 

The stable and unstable invariant cylinders intersect this section infinitely often, preserving each area bounded by the closed curve of and IT, although it will become indefinitely deformed due to their homoclinic tangles. However, one of the most striking consequences deduced from the analyses of the initial intersection of the invariant cylinder manifolds at a certain Poincare section defined in region A is this If and only if the system lies in the interior of 11 11a, the system can climb through from A to B whenever wandering in the... 

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. 

As I decreases, the stable limit cycle moves down and squeezes U closer to the stable manifold of the saddle. When / = /<., the limit cycle merges with t/ in a homoclinic bifurcation. Now f/ is a homoclinic orbit—it joins the saddle to itself (Figure 8.5.8). 

Scaling near a homoclinic bifurcation) To find how the period of a closed orbit scales as a homoclinic bifurcation is approached, we estimate the time it takes for a trajectory to pass by a saddle point (this time is much longer than all others in the problem). Suppose the system is given locally by x x, y = -2, y. Let a trajectory pass through the point (/i,l), where f.i 1 is the distance from the stable manifold. How long does it take until the trajectory has escaped from the saddle, say out to x t 1 (See Gaspard (1990) for a detailed discussion.)... 

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

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. 

For systems of n degrees of freedom, take a normally hyperbolic invariant manifold with 2r normal directions in phase space. Note that for Hamiltonian systems, the dimension of the normal directions in phase space is alway even, because the eigenvalues of the variational equation (Jacobi equation) is symmetric around the value 0. Thus, the dimension of the normally hyperbolic invariant manifold is 2n — 2r and, for its stable and unstable manifolds, their dimensions are 2n — r, respectively. The dimension of their homoclinic intersection, if it exists, is 2n — 2r in the 2n-dimensional phase space. When we consider the intersection manifold on the equi-energy surface, its dimension on the surface is 2n — 2r — 1. Thus, the dimension d of the intersection on the Poincare section is d = 2n — 2r — 2. 

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. 

In Figure 3.11, a typical case of homoclinic tangency is indicated by showing those sections of the stable and unstable manifolds. Here, we can... 

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. 

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