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Local stable invariant manifold

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... [Pg.21]

This is also the Hamiltonian of the activated complex. We will encounter it in Eq. (23) with the customary symbol H. Regardless of its stability properties or the size of the nonlinearity, Eq. (12) is always an invariant manifold. However, we are interested in the case when it is of the saddle type with stable and unstable manifolds. If the physical Hamiltonian is of the form of Eq. (1), then a preliminary, local transformation is not required. The manifold (12) is invariant regardless of the size of the nonlinearity. Moreover, it is also of saddle type with respect to stability in the transverse directions. This can be seen by examining Eq. (1). On qn = Pn = 0 the transverse directions, (i.e., q and p ), are still of saddle type (more precisely, they grow and decay exponentially). [Pg.187]

This enables us to extract and visualize the stable and unstable invariant manifolds along the reaction coordinate in the phase space, to and from the hyperbolic point of the transition state of a many-body nonlinear system. PJ AJI", Pj, Qi, t) and PJ AJ , Pi, q, t) shown in Figure 2.13 can tell us how the system distributes in the two-dimensional (Pi(p,q), qi(p,q)) and PuQi) spaces while it retains its local, approximate invariant of action Jj (p, q) for a certain locality, AJ = 0.05 and z > 0.5, in the vicinity of... [Pg.110]

Because the perturbation is Hamiltonian, the 3D level energy surfaces are preserved. In the 4D normally hyperbolic invariant manifold of the unperturbed space, the locally stable and unstable manifolds and the flow describe the geometric structure of the perturbed phase space given by the perturbed normally hyperbolic locally invariant manifold, the locally stable and unstable manifolds, and the persistence of the 2D nonresonant invariant tori T-,(Pi,P2)-... [Pg.81]

On M-y there are locally stable and unstable manifolds that are of equal dimensions and are close to the impertm-bed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold intersects each of the 5D level energy sm-faces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem. The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. [Pg.81]

Most of the 2D nonresonant invariant tori T(Pi,P2)) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold and are KAM tori. In the phase space of the perturbed system 7 > 0 and a = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies conditionally-periodic motions of the perturbed system are smooth functions of the perturbation 7. A generahzation of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds, b fn order to determine if chaos exists, two measurements are required in order to determine whether or not and VK (T.y) intersect transversely. [Pg.82]

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... [Pg.83]

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. [Pg.309]

Let us straighten the local stable and unstable invariant manifolds. Then the system near O takes the form (see Sec. 2.7)... [Pg.322]

In the original proof the system under consideration was assumed to be analytic. Later on, other simplified proofs have been proposed which are based on a reduction to a non-local center manifold near the separatrix loop (such a center manifold is, generically, 3-dimensional if the stable characteristic exponent Ai is real, and 4-dimensional if Ai = AJ is complex) and on a smooth linearization of the reduced system near the equilibrium state (see [120, 147]). The existence of the smooth invariant manifold of low dimension is important here because it effectively reduces the dimension of the problem. ... [Pg.380]

Let p = x(0) be an arbitrary initial point with Jr,(0) > 0. Then the initial data do not belong to either stable manifold. Hence w p) is not equal to either Eq or E2, but it does lie on E = 0. Since it is invariant and since every solution of (5.2) on E = 0 converges to an equilibrium, u(p) contains an equilibrium. By the Butler-McGehee theorem, Eq uIp) since M (Eq) is unbounded. If o p) contains E2, then o p) also contains either Eq or an unbounded orbit, again by the Butler-McGehee theorem (see Figure 5.2). Since this is impossible, E must be in 0 p). However, is a local attractor, so u p) = Ey This completes the proof. ... [Pg.18]

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. [Pg.9]


See other pages where Local stable invariant manifold is mentioned: [Pg.141]    [Pg.141]    [Pg.12]    [Pg.271]    [Pg.147]    [Pg.167]    [Pg.555]    [Pg.695]    [Pg.74]    [Pg.317]    [Pg.82]    [Pg.111]    [Pg.118]    [Pg.320]    [Pg.378]    [Pg.134]    [Pg.297]   
See also in sourсe #XX -- [ Pg.509 ]




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