Closed invariant curve

A non-resonant invariant curve survives as closed invariant curve, due to the KAM theorem (see Figure 14b). 

The smooth case corresponds, in particular, to a small time-periodic perturbations of an autonomous system possessing a homoclinic loop to a saddle-node equilibrium (see the previous section). Indeed, for a constant time shift map along the orbits of the autonomous system the equilibrium point becomes a saddle-node fixed point and the homoclinic loop becomes a smooth closed invariant curve, but the transversality of to F is, obviously, preserved under small smooth perturbations. 

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

It is possible, however, to obtain generalized concentration-invariant curves under straining rates close to those used in the real-scale commercial process. In [163, 164, 209] the generalized curves for PE based composites were obtained by the procedure described in [340] by carrying out nonrotationa shifts in the vertical and horizontal directions the authors sought to achieve the closest coincidence between the experimental curves in the lg t] — lg x coordinates for the base polymer and the curves for filled composites. 

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... 

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. ... 

As shown in Figure 3.5.3, the relaxation time versus pressure curves are dramatically different from those obtained using CF4 at a temperature well above its critical point. Indeed, while the overall form of the Tx curves for CF4 in fumed silica was similar to that of the bulk gas, the shape of the Ti plots for c-C4F8 in Vycor more closely resembles that of an adsorption isotherm (Ta of CF4 in Vycor is largely invariant with pressure, as gas-wall collisions in this material are more frequent than gas-gas collisions). This is not surprising given that we expect the behavior of this gas at 291 K to be shifted towards the adsorbed phase. The highest pressure... 

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. 

Choose a parameterization for the invariant closed curve. In benign cases when the curve is not highly irregular this can be accomplished by... 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

The regions of existence and appearance of reference-invariant functions % (u, i j) are represented in Fig. 10. Curves with maxima and minima cannot be described in a reference-invariant manner. In this case, both the dimensional-analytical representation and the model material system are confined to the region close to the standardization range . 

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... 

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