Stable manifolds

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

Pi = Qi = 0 as t —y oo, or, more precisely, they approach the NHIM of the appropriate energy within the central manifold. Due to this behavior, the set of all initial conditions with Q = 0 is called the stable manifold of the NHIM. Similarly, trajectories with Pi = 0 asymptotically approach the NHIM as t > —oo. They are said to form the unstable manifold of the NHIM. 

In the absence of damping (and in units where ( b = 1), the invariant manifolds bisect the angles between the coordinate axes. The presence of damping destroys this symmetry. As the damping constant increases, the unstable manifold rotates toward the Agu-axis, the stable manifold toward the A<7u-axis. In the limit of infinite damping the invariant manifolds coincide with... 

The most important of these manifolds for the purposes of TST are, as before, the surface given by AQi = APi that serves as a recrossing-free dividing surface and the stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories. The latter are given by AQi = 0 (stable manifolds) or APj = 0 (unstable manifolds), respectively. Together, they can be characterized as the zeros of the reactive-mode action... 

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. 

Now we shall consider the general case. As remarked before, we may dehne the stable manifold IF+ and the unstable manifold W by using the C -action, i.e. 

S Yj is Lagrangian, the normal bundle is identified with T (S Y). Moreover T S Y) is dense in since it is identified with the stable manifold of S Y. Hence we may say... 

Let us denote by Wj (resp. by W ) the stable manifold of Cv (resp. the unstable manifold of Cv). These are defined by... 

This shows that As looks roughly like T Als- These are not exactly the same since may not be stable for ( , ) e As- On the other hand we have shown that C T A Since SnT, is Lagrangian, the normal bundle is identified with T (5n ). Moreover T (5n ) is dense in T since it is identified with the stable manifold of Hence we may say that t Y looks like Sn(T Y,) but not exactly. 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

An example is shown in figure 3 for section AA near the bottom of the 2/1 resonance horn of figure 2. As the frequency is increased from left to right, the torus becomes phase locked as a pair of period 2 saddle nodes develop on it. The saddle nodes then separate with the saddles alternating with the node and the invariant circle is now composed of the unstable manifolds of the saddles whereas the stable manifolds of the saddles come from the unstable period 1 focus in the middle of the circle and from infinity. As the frequency is increased further, the saddles rotate around the circle and recombine with their neighbouring nodes in another saddle-node bifurcation. 

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. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

The saddle cycles L and L2 surround the stable states P and P2 and are located at the intersection of the unstable Wu and stable Ws manifolds. The unstable manifold goes to the stable state P from one side and to the chaotic attractor from the other side. The stable manifold Ws forms a tube in the vicinity of the stable state [183]. The saddle cycles L and L2 have the multipliers (1.0000,1.0280,0.0001), and therefore trajectories will go slowly away along the unstable manifold, and they will approach quickly along the stable manifold. 

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... 

The basic picture discussed above is quite general in Hamiltonian systems. Of particular importance is the concept of stable and unstable manifolds associated with unstable periodic orbits. Trajectories along the stable manifold will be mapped toward the periodic orbit, whereas trajectories along the unstable manifold will be mapped away from the periodic orbit. It turns out that the union of segments of the stable and unstable manifolds is very useful in defining the reaction separatrix and calculating the flux crossing the separatrix in few-dimensional systems. 

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. 

The NHIM and Its Stable and Unstable Manifolds. In the normal-form coordinates the NHIM is given by q = = 0, the stable manifold of the... 

The local and global stable unstable manifold theorems (see, e.g.. Ref. 24, pp. 136-140) tell us the following are the (un)stable manifolds) ... 

Figure 11. Schematic view of a TS (thick black line), with the same type of view as in Fig. 10. The equilibrium point is in the middle with its stable manifold and unstable manifold extending as straight lines. Trajectories in dot-dashed lines are reactive (inside the tubes) and cross TS trajectories in dashed lines are not reactive. The whole gray surface is the energy level. For a linear motion, it takes the form of a parabolic hyperboloid.

Furthermore, we investigate the detailed structure of the Poincare surface of section for the case of (Z, ) = (1,1). In this case, there are tori. These tori have the periodic points with period 6 in their outermost part. Here we counted the number of vertices of two triangle, namely 2x3 = 6. These periodic points is associated to one orbit in the whole phase space, which is an antisymmetric orbit in the configuration space. These periodic points have stable and unstable manifolds. In Fig. 8a, we depict the stable manifolds of the these periodic points. In Fig. 8b, we also depict the unstable manifolds by using the symmetry. The stable and unstable manifolds of these periodic points go to It should be noted that the reached points of them on 0 is the accumulation points of and Comparing Figs. 6a and 6b with Fig. 8, it is confirmed that in Figs. 6a and 6b is nearly parallel to the stable manifolds and in Fig. 6b is nearly parallel to the unstable manifolds. Therefore, it is understood that the foliated structure of tc and manifests the foliation of the stable and unstable manifolds—that is, hyperbolic structure. 

Figure 8. The stable and unstable manifolds of the periodic points of the outermost tori for the case of (Z, 5) = (1,1). (a) The stable manifolds, (b) The unstable manifolds are also added to (a).

In constructing the stable manifold of the NHIM we follow, backward in time, the normal directions of the NHIM with negative local Lyapunov exponents. For the unstable manifold we follow forward in time the normal directions of the NHIM with positive local Lyapunov exponents. 

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