Smooth invariant manifold

Originally, this theorem was proved for C -smooth systems. We stress here that our proof includes the Crease which allows for a direct use of this theorem in the situation where the system is defined on a -smooth invariant manifold (see Theorem 13.9). 

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. ... 

The corresponding structure of fast-slow time separation in phase space is not necessarily a smooth slow invariant manifold, but may be similar to a "crazy quilt" and may consist of fragments of various dimensions that do not join smoothly or even continuously. 

Fenichel, N., 1971, Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. 

Investigation of the vicinity of smooth invariant toroidal manifold for the... 

The important results concerning the location of integral curves of systems of nonlinear equations in the vicinities of smooth toroidal invariant manifold and compact invariant manifold were also obtained by use of the method of rapid convergence (Bogolyu-bov, Mitropolsky, and Samoilenko, 1969 Samoilenko, 1966a,b). 

Let be a four-dimensional smooth symplectic manifold on which a Hamiltonian system v = sgrad H is given, with H being a smooth Hamiltonian. Equilibrium positions xq of the system v are critical points of the function H. Since fT is an integral of the system u, it follows that the field v may be restricted to an invariant three-dimensional constant-energy surface Q, that is, Q = x M H x) = const. Being a symplectic manifold, is orientable, and therefore the manifold Q is abo orientable. 

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. 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

As shown in Chap. 5, the above critical equilibrium state lies in an invariant C -smooth center manifold defined by an equation of the form y = (a ), where (x) vanishes at the origin along with its first derivative. 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

Let us now consider the question concerning what happens when is non-smooth. For the first time, this question was studied in [3] where it was discovered that the possibility of the breakdown of the invariant manifold causes an onset of chaos at such bifurcations. In particular, sufficient conditions (the so-called big lobe and small lobe conditions) were given in [3] for the creation of infinitely many saddle periodic orbits upon the disappearance of a saddle-node in the non-smooth case. Subsequent studies have shown that these conditions may be further refined so we may reformulate them as follows. 

Let us consider next the bifurcation of the saddle-node periodic orbit L in the case where the unstable manifold is a Klein bottle, as depicted in Fig. 12.3.1, i.e. when the essential map has degree m = -1. By virtue of Theorem 12.3, if is smooth, then a smooth invariant attracting Klein bottle persists when L disappears. In its intersection with a cross-section So, the flow on the Klein bottle defines a Poincare map of the form (see (12.2.26))... 

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]. 

When the separatrix returns to O, it lies in the stable manifold y = 0. If the system has order more then three, we will assume that F does not belong to the strong stable manifold Recall that is a smooth invariant... 

Note that at // = 0 and ry 0, the separatrix Fi forms a homoclinic loop, approaching one of the two components of Wj c ioc depending on the sign of Tj, Since the non-degeneracy conditions of Theorem 13.7 are satisfied for 17 7 0, the Poincare map T has a smooth invariant curve through the point M" "(0,7 , li" ), transverse to the stable manifold. When restricted to this curve, the map T assumes the form... 

Note that all of these results (except for the subtle structure of the set of curves C12 in the case where Oi is a saddle-focus and O2 is a saddle) are proven for C -smooth systems. Therefore, just like in the case of a homoclinic-8, these results can be directly extended to the case where the unstable manifolds of Oi and O2 are multi-dimensional (but they must have equal dimensions in this case), provided that the conditions of Theorem 6.4 in Part I of this book, which guarantee the existence of an invariant -manifold near the heteroclinic cycle, are satisfied. 

Consider a C -smooth (r > 3) system in a neighborhood of a saddle equilibrium state with m-dimensional stable and n-dimensional unstable invariant manifolds. 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Using graph transforms, Ad automatically becomes time- and flow-invariant. Smoothness of the manifold Ad is a very delicate question and has been settled in [33]. 

Existence of Invariant Toroidal Manifolds With Loss of Smoothness... 

The question of the construction of a nontrivial smooth SC-metric on S was solved in 1902 by Zoll [190] who explicitly constructed a real-analytic SC rotation surface homeomorphic to 5. A simple generalization of the Zoll construction leads to constructing a family of smooth 5C-metrics on S dependent on the functional parameter and invariant under rotations around a certain axis [I9lj. In his book [192] Blaschke proposed an exquisite modification of the Zoll construction which allows us to construct 5C-manifolds which are homeomorphic to the sphere but are not rotation surfaces. Blaschke surfaces are glued of pieces of different SC rotation surfaces by means of films isometric to parts of the standard sphere. 

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