Stable invariant curve

It follows from the annulus principle (Theorems 4.2 and 4.5) that there exists a smooth stable invariant curve C of the form... 

The above theorem is related to the map on the center manifold. Reconstructing the behavior of trajectories of the original map (11.6.2) is relatively simple. Here, if L < 0, then the fixed point is stable when /i < 0. When /i > 0 it becomes a saddle-focus with an m-dimensional stable manifold (defined by T = 0) and with a two-dimensional unstable manifold which consists of a part of the plane y = 0 bounded by the stable invariant curve C,... 

Note that both the saddles and nodes appearing inside the resonant wedge (called Arnold tongue, sometimes) lie on the invariant curve (stable if Li < 0 or unstable if L >0). Since the only stable invariant curve that can go through a saddle is its unstable manifold, and since the only unstable curve that can also go through a saddle is its stable manifold, it follows that inside the resonance zone the invariant curve is the union of the separatrices of saddles (imstable separatrices if Li < 0, or stable separatrices if L > 0) that terminate at the nodes. 

Thus, when a and are small, the sign of the first Lyapunov value equals the sign of the difference (/ — 2a). If it is negative, the stable invariant curve is born through the super-critical Andronov-Hopf bifurcation when crossing the curve AH towards larger (3. ... 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

This curve has two branches, m + v = 0 and m — v = 0, or in the original coordinates, x = 0 and j = 0, that is, there is no adsorbed oxygen or carbon monoxide present on the catalyst surface. Both branches of the slow invariant curve are stable. The two branches have an intersection point (0,0), which is definitely not a steady-state point of the original system. Therefore, around this point a more accurate approximation is necessary. 

If the first Lyapunov value Li > 0, then the fixed point of the map (11.6.6) is unstable for sufficiently small /x > 0. When /x < 0 the fixed point is stable its attraction basin is the inner domain of an unstable smooth invariant curve of the form (11.6.7). As p —0, the curve collapses into the fixed point see Fig. 11.6.2). 

If Li > 0, then when /i > 0, the fixed point is a saddle-focus of the above type, but its unstable manifold is the whole plane y = 0. Upon entering the region M < 0, the fixed point becomes stable. Meanwhile a saddle invariant curve C bifurcates from the fixed point its unstable manifold is (m -h 1)-dimensional and consists of the layers x — constant, restored at the points of the invariant curve. The stable manifold separates the attraction basin of the point O all trajectories from the inner region tend to O, and all those from outside of Wq leave a neighborhood of the origin. 

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 11.4 shows essentially that outside the narrow sector bounded by 1 and 2, the bifurcation behavior does not differ from that of equilibrium states (see Sec. 11.5) fixed points correspond to equilibrium states, and the invariant curves correspond to periodic orbits. However, the transition from the region D2 to the region Dq occurs here in a more complicated way. In the case of equilibrium states the regions D2 and Do are separated by a line on which a stable and an imstable periodic orbits coalesce thereby forming a semi-stable cycle. In the case of invariant closed curves, the existence of a line corresponding to a semi-stable invariant closed curve is possible only in very degenerate cases (for example, when the value of R does not depend on as... 

The boundary of the resonant zone corresponds to a coalescence of the stable and unstable periodic orbits on the invariant circle, i.e. to the saddle-node bifurcation of the same type we consider here. Besides, if there were more than two periodic orbits, saddle-node bifurcations may happen at the values of parameters inside the resonant zone. By the structure of the Poincare map (12.2.26) on the invariant curve,... 

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... 

When 6=1, the Henon map becomes conservative, as its Jacobian equals -1-1. At 6 = 1 and a = —1, it has an unstable parabolic fixed point with two multipliers +1 at 6 = 1 and a = 3, it is a stable parabolic fixed point with two multipliers —1. In between these points, for —1 < a < 3 (i.e. (a, 6) G T), the map has a fixed point with multipliers where cos > = 1 y/a -h 1. This is a generic elliptic point for tp 7r/2,27r/3,arccos(—1/4) [167]. Since the Henon map is conservative when 6=1, the Lyapunov values are all zero. When we cross the curve AH, the Jacobian becomes different from 1, hence the map either attracts or expands areas which, obviously, prohibits the existence of invariant closed curves. Thus, no invariant curve is born upon crossing the curve AH. ... 

Such a scenario of stability loss is often referred to in the literature as soft (see Chap. 14). In the case Li > 0, the loss of stability develops in a dangerous way the point O is stable initially meanwhile an unstable invariant curve materializes from the homoclinic tangles of 0, and shrinks to the origin as the curve AH is reached. The fixed point at the origin becomes unstable upon crossing AH and all nearby trajectories escape from its neighborhood. ... 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

Fig. 6.6 shows equilibria in the CaO-AljOj-HjO system mainly as found by Jones and Roberts (J17) at 25°C and reported by Jones (J19), who also reviewed earlier studies. Table 6.6 gives invariant concentrations at 20°C. The stable phases in contact with solution at 25°C are gibbsite (AHj), C3AH5 and CH. but metastable solubility curves for AFm phases, CAHjo and hydrous alumina can also be obtained. 

The stable and unstable invariant cylinders intersect this section infinitely often, preserving each area bounded by the closed curve of and IT, although it will become indefinitely deformed due to their homoclinic tangles. However, one of the most striking consequences deduced from the analyses of the initial intersection of the invariant cylinder manifolds at a certain Poincare section defined in region A is this If and only if the system lies in the interior of 11 11a, the system can climb through from A to B whenever wandering in the... 

As for monotropic polymorphism, the common L V curve will normally intersect the Si V and -S n-F curves below their intersection (Figure 3) (4). There is no region of stability for the second polymorph (-S ), and the melting point of the metastable An polymorph will invariably be lower than that of the stable form (Ai). Unlike enantiotropic polymorphism, the triple point is always higher than the melting point of the stable 5i phase. Only one of the polymorphs remains stable up to the melting point upon heating, and the other polymorph can exist only as a metastable phase, irrespective of... 

Stability. As long as the temperature remains below Tg, the composition of the system is virtually fixed. This implies physical stability crystallization, for instance, will not occur. As mentioned, some chemical reactions may still proceed, albeit very slowly because of the high viscosity and the low temperature. The parameters Tg and i//s are, however, not invariable they are not thermodynamic quantities. Their values will depend to some extent on the history of the system, such as the initial solute concentration and the cooling rate. The curve in Figure 16.6 denoted rff (for fast freezing) shows what the relation may become if the system is cooled very fast. The Tg curve is now reached at a lower ice content, so the apparent Tg and i// values are smaller. However, the system now is physically not fully stable water can freeze very slowly until the true i// s is reached. 

On following the solubility curve of the hexahydrate from the ordinary temperature upwards, it is seen that at a temperature of 29 8° represented by the point H, it cuts the solubility curve of the a-tetrahydrate. This point, therefore, represents an invariant system in which the three phases hexahydrate, a-tetrahydrate, and solution can coexist under constant pressure. It is also the transition point for these two hydrates. Since, at temperatures above 29 8°, the a-tetrahydrate is the stable form, it is evident from the data given before (p. 184), as also from Fig. 79, that the portion of the solubility curve of the hexahydrate lying above this temperature represents metastable equilibria. The realisation of the metastable melting-point of the hexahydrate is, therefore, due to suspended transformation. At the transition point, 29 8°, the solubility of the hexahydrate and a-tetrahydrate is 100 6 parts of CaClg in 100 parts of water. 

The KAM theorem, it should be noted, has nothing to say about what happens when the strength of the perturbation increases. However, a considerable amount of experience has accumulated from detailed numerical calculations performed for many systems. One can visualise the results by studying Poincare sections if a cut is made across an invariant torus (see fig. 10.3) and a numerical calculation of trajectories is performed over a sufficiently long time, the stable orbits fill the deformed tori densely, and so result in closed curves in the two-dimensional cut, whereas the irregular or chaotic orbits yield a random speckle. 

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