Smooth invariant curve

All consecutive points of the map lie on smooth invariant curves that are circles with radii g = s/2J. The rotation angle A0 as well as the ratio varies along the radius g (see Figure 14a). 

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). 

Theorem 11.6. Let a smooth annulus map have a smooth invariant curve and let the rotation number on the invariant curve he irrational, Then by an arbitrarily small smooth perturbation infinitely many periodic orbits may be bom. 

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... 

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

Remark. Since the smoothness of the invariant curve cannot, in general, exceed the smoothness of the map itself, the function in (11.6.7) is only C -smooth with respect to (p and 6 -smooth with respect to fx. In fact, the... 

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. 

Observe that near each node point the invariant curve coincides with its leading manifolds. It follows that the invariant curve has a finite smoothness, generally speaking. 

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... 

Figures 13.4.6 Ap > 0) and 13.4.7 Ap < 0) show how the image of 5 by the map T moves when the loop is split. In any case, since the map T is contracting in the a-variables and expanding in y, it follows that it has a smooth attracting invariant curve lo p), transverse to yo = 0 in 5q.

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. 

All such points formed smooth curves when plotted on the three-component triangular graphs used for each system investigated except that of the potassium chloride-water-THF system. In the latter case, two solubility curves appeared which met in an invariant point. The solubility plots are shown in Figures 1, 2, and 3. 

From the point of view of general methodology, several comments are in order. First, the appearance of the Fourier-Bessel transform in the stmcture function [Eq. (20)] reflects on the breakdown of translational invariance, which is prevalent in the case of the bulk. Second, the different symmetries of spherically projected structure functions for the finite system and of plane wave structures for the bulk system are crucial for a proper representation of the cluster excitations. Third, the discrete eigenvectors k n are determined by the boundary conditions. Fourth, the energies kin) are discrete. However, the complete spectrum for a fixed value of n containing 1 = 0, 1, 2,... branches would form a continuous smooth curve. 

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). 

Considering capillary dynamics, the pressure drop term is often described by the Laplace equation, AP = 2yH, where y represents liquid surface tension and H represents the mean curvature of the liquid-gas interface associated with aU curves, C, passing through the surface. Furthermore, the character of a sufficiently smooth surface is through the invariant from differential geometry, the principal curvature of each curve, kj. The radii of curvature are the inverse of each principal curvature, A , = 1/r,. Considering the maximum and minimum radii of curvature at a point on a three-dimensional surface, the mean curvature can be calculated explicitly (see Appendix for more thorough derivation of the mean curvature parameter) ... 

Failure to produce a smooth curve with a well-defined point of inflection is almost invariably due to a defect in the membrane or its attachment to the cap, or to a faulty electrical connection. 

Probably these regions correspond to the second phase - MgAl204, defined by X-ray. It is proven by the combined structural analysis in "height" and "mag-cos" modes (figure 10). Clear peaks on the "mag-cos" curve (fig. 10, c) and corresponding smooth relief contour show the invariation of the block surface elastic properties and the presence of the doped... 

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