Local invariant manifold

For systems with lag, these problems have been investigated less satisfactorily. Indeed, though a series of results has been obtained up to now in the theory of local invariant manifolds for systems with lag by V.Fodchuk (1965, 1970), A.Halanay (1965), J.Hale (1966), Yu.Neimark (1975), A.Zverkin (1970), and others, there exists a comparatively small number of works devoted to the study of invariant toroidal manifolds for these systems (Martinyuk and Samoilenko, 1974a, 1976 Ordynskaya, 1976,1977,1979). 

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)-... 

Persistence of M. The perturbed system 7 > 0 and a = 0 possesses a 4D normally hyperbolic locally invariant manifold M, given by... 

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. 

KAM TheoremP The KAM theorem determines whether the recurrent motions occm- on the pertm-bed normally hyperbolic locally invariant manifold M. and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian pertm-bation. The unpertm-bed Floquet Hamiltonian (P)(7 = 0) = Po(p,PljPj) + He(q,P,Pi,P2) satisfies the following nondegeneracy (or noiu eso-nance) condition ... 

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. 

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

A bundle is a structure consisting of a manifold E. and manifold M, and an onto map it E M. 6Symplectic topology is the study of the global phenomena of symplectic symmetry. Symplectic symmetry structures have no local invariants. This is a subfield of topology for an example, see McDuff and Salamon [14],... 

In summary, the principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. The electromagnetic field is the compensating field of local phase transformation and the gravitational field is the compensating field of local Lorentz transformations. 

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 ,... 

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). 

First, in order to simplify the description of the dynamics we separate the whole system, locally in the phase space, into two parts based on a gap in characteristic time scales. This is done using the concept of normally hyperbolic invariant manifolds (NHIMs) [4-8]. Here, the characteristic time scales are estimated as the inverses of the absolute values of the local Lyapunov exponents [5,6]. Then, the Fenichel normal form offers a simplified description of the local dynamics near a NHIM [7]. 

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... 

The observed scaling of p has an important physical interpretation Once p approaches the characteristic invariant and statistical distributions generated by the global invariant manifold, it then evolves everywhere at the same rate as the mean density. In other words, if the mean intermaterial area density is doubled, the local density is doubled everywhere. This is important, because it means that the time evolution of time evolution of p at aU locations of the chaotic flow (i.e., intimacy of mixing improves everywhere by the same factor). Similarly, the striation thickness both locally and globally ... 

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]. 

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

Let the separatrix Fi" enter the saddle from the half-plane x > 0, and let the separatrix Tf leave O towards positive values of y. Since we have not yet straightened the invariant manifolds, the local equations of and F are given by, respectively,... 

On the cross-section 5o, the local unstable manifold of the fixed point M (/x) is a small piece of the invariant curve Zo(/x) through this point. The entire imstable manifold of M (/x) on Sq can be obtained by iterating the local unstable manifold under the action of the map T. Since the domain of the map T is bounded by the surface yo = 0 = fl So, the unstable... 

In these coordinates the invariant manifolds are locally straightened their equations are = x = 0,u = 0 and = y = 0, respectively. 

In dimensions higher than three, the condition A 0 is an essential nondegeneracy condition. It is important that we use the coordinates in which the system has locally the form (13.4.10) and that the identities (13.4.11) are hold. In these coordinates the intersection of with Sq is the straightline yo = 0, and the intersection of the extended unstable manifold with Si is tangent to the space m = 0 (the extended imstable manifold is a smooth invariant manifold which is transverse to at O). Thus, one can see from (13.4.14) that the condition A 7 0 is equivalent to the condition of transversality of n5i) to at the point M+, i.e. to the transversality condition... 

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. ... 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

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