Local stable manifold

Proof. Assertion (1) is just Theorem C.4. The assertion concerning M (xf) follows from the Perron-Frobenius theory (Theorem A.5) and the monotonicity of the time-reversed system (6.3). If J is the Jacobian matrix of / at Xq, then (6.2) implies that —J satisfies the hypotheses of Theorem A.5. It follows that r = —s —J) < 0 is an eigenvalue of J corresponding to an eigenvector u > 0. Because M (Xo) is tangent at Xq to the line through Xq in the direction v, the local stable manifold of Xq is totally ordered. Since M X()) is the extension of the local stable manifold by the order-preserving backward (or time-reversed) system, it follows... 

As /i —0, the stable manifold of L fx) approaches the component of W 0) W 0) which contains F. The characteristic form of in the small neighborhood U of F is sketched in Fig. 13.4.11. Here, the local stable manifold can be continued along F in backward time so that it returns to the small neighborhood of O. Since A 0, the manifold is transverse to and therefore it is limited to the manifold as t —> — oo. This geometry of is a direct consequence of the non-vanishing of the value of A. We remark, however, that in some cases, even if A = 0, the manifold may be still limited to in backward time. 

Proposition 1. For e sufficiently small the periodic orbit 7(t) of (7) survives as aperiodic orbit 7 (t) = rj(t)+0(e), of (10) having the same stability type as 7(t), and depending on e in a C2 manner. Moreover, the local stable and unstable manifolds Wlsoc( ye(t)) and Wfoc e(t)) of 7 (t) remain also e-close to the local stable and unstable manifolds WfoMt)) and Wfioc( (t)) of ft), respectively. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

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) ... 

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. 

Since the manifold Mq is a NHIM, it changes continuously, under a small perturbation, into a new NHIM M - Moreover, the separatrix Wq changes, continuously and locally near M, into the stable manifold and the unstable one W" of the NHIM M. Note, however, that, in general, and W no longer coincide with each other to form a single manifold globally. Then, the Lie transformation method brings the total Hamiltonian H x.I, 0) into the Fenichel normal form locally near the manifold M. ... 

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. ... 

The behavior of orbits of P near a fixed point x can be described in the case where x is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at x has modulus equal to 1. In this case there exist (local) stable and unstable manifolds M (x) and M (x) (respectively) containing the point x which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at x. (The stable (unstable) subspace... 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

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.)... 

Recall that the roll pattern becomes stable for T > Ti = 4/3. Hence, in the interval Ti < T < Ts the Lyapunov function has 4 local minima, three of them correspond to three types of roll patterns, and one of them corresponds to hexagons. The basins of attractions between them are separated by stable manifolds of some additional saddle-point stationary solutions, corresponding to squares (e.g. R = R2 0, Rs = 0) and skewed hexagons" (e.g. R = R2 7 R3 7 0). Finding the latter solutions is suggested to the readers as an exercise. 

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)-... 

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. 

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 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... 

The formal analogy with a velocity field (i.e. W(r) = dr/dt) enables to build trajectories by integrating over the time variable. Each trajectory starts in the neighborhood of a point (or set of points) called the a-limit for which W(r) =0 and ends in the neighborhood of another point (or set of points) called the (u-limit for which also W(r) = 0. Except for asymptotic behaviors, the a and cu-limits are critical points. The set of trajectories having a given critical point as (u-limit is called the stable manifold of this critical point whereas its unstable manifold is the set of trajectories for which it is a a-limit. The stable manifold of a critical point of index 0 (a local maximum or attractor) is the basin of the attractor, that of a critical point of index larger than 0 is a separatrix it is the boundary between basins. 

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. 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

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

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. ... 

Applications of isotopes in biochemistry have been manifold. Besides tracing pathways and following the rates of isotope uptake and their intracellular localization, labeling molecules in selected stable positions allowed studies to be made of reaction details, such as those in citrate... 

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