Invariant curves

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. 

We are now concerned with the location of two-tori in our systems. We have formulated for that purpose a shooting on collocations iterative algorithm that computes an invariant curve of the stroboscopic map for a forced system. Part of this algorithm and its application to maps in general are described elsewhere (Kevrekidis et al., 1985). We briefly describe here its application to a two-dimensional forced system. 

Chenciner, A. 1985 Hamiltonian-like phenomena in saddle-node bifurcations of invariant curves for plane diffeomorphisms. In Singularities and dynamical systems (ed. S. N. Puevmat-ikos). Amsterdam Elsevier Science Publishers/North Holland. 

The action of the sealing invariant curves can be demonstrated directly with the help of a Monte-Carlo calculation. Consider an ensemble of M rotors with initial conditions = 0 and 9 = 2(m —... 

The SSE phase-space portrait shown in Fig. 6.5 reminds us of the phase-space portraits of the kicked rotor presented in Chapter 5. In Fig. 6.5 we can identify resonances and sealing invariant curves. In Chapter 5 we saw that resonance overlap in the standard mapping defines a sudden percolation transition when for K > Kc the seahng invariant... 

A Poincare surface of section may be used to identify the chaotic and quasi-periodic regions of phase space for a two-dimensional Hamiltonian. An ensemble of trajectories, chosen to randomly sample the phase space, are calculated and for each trajectory a point is plotted in the (9i,Pi)-plane every time Q2 = 0 for p2 > 0. A quasi-periodic trajectory lies on an invariant curve, while the points are scattered for a chaotic trajectory with no pattern. Figure 44 shows an example for a two-dimensional model for HOCl the HO bond distance is frozen in these calculations [351]. It clearly illustrates how the phase space becomes gradually more chaotic as the energy increases. 

Moser, J. (1962). On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gott,. II Math. Phys. Kl 1962, 1-20. 

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). 

All points on a resonant invariant curve (circle) are r-multiple fixed points the point comes to the initial position after r rotations along the angle 2 on the 2-torus. (A i = 2irs and A 2 = 2ttv). It can be easily seen that the unperturbed mapping (73) can be obtained from the generating function... 

What happens to the unperturbed circular invariant curves when e > 0 ... 

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

Figure 14- (a) The resonant and non resonant invariant curves of the integrable... 

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. 

All invariant curves on the X Y plane are circles with constant radii y/2Ji, where J = /Gma — J20 It is clear that the radius of the invariant curve depends on a fixed value of the semimajor axis a. 

On a particular invariant curve, with radius y/2J, there corresponds a certain eccentricity, obtained from J = y/Gma 1 — Vl - e2). 

The resonant invariant curves correspond to the resonant elliptic periodic orbits, in the rotating frame. 

A resonant n/n = p/q elliptic periodic orbit is a multiple fixed point on the resonant invariant curve. The angle i) changes during one iteration by... 

All points on this invariant curve are fixed points. A linear analysis shows that the two eigenvalues of a fixed point are equal to 1. This means that the periodic elliptic orbits in the rotating frame have two pairs of unit eigenvalues. 

We remark that all the fixed points on a resonant invariant curve of the Poincare map correspond to elliptic motion of the small body, with the same semimajor axis a, such that n/n is rational, and the same eccentricity e. They differ only in the orientation, which means that all these orbits have different values of w, as shown in Figure (15). 

According to the KAM theorem (Guckenheimer and Holmes, 1983), for sufficiently small e, the non resonant invariant circles survive the perturbation as nearly circular invariant curves. These invariant curves represent nearly elliptic orbits of the small body that are not periodic both in the rotating frame and the inertial frame. 

Olivera, A. and Simo, C. (1987). An obstruction method for the destruction of invariant curves. Physica D, 26 181. 

Phase space curves, associated with a one-dimensional attractive potential such as that for He-I, are illustrated in figure 8.12. The separatrix (i.e., dashed curve) is called a reaction separatrix, because it is a boundary between bound unreactive motion and unbound reactive motion. Regardless of the length of time the trajectory is evaluated it will remain on this curve and, therefore, the separatrix is called an invariant curve. It is plotted in figure 8.13(a). 

If the He-I one-dimensional potential is coupled to other degrees of freedom as in Hel2, the separatrix is no longer an invariant curve (Davis and Gray, 1986 Davis and Skodje, 1992). Trajectories that initially do not have sufficient energy in the He-I coordinate to cross the separatrix and dissociate can acquire the needed energy by... 

The molecular theory predicts strong temperature dependenee of the relaxation ehar-acteristics of polymeric systems that is described by the time-temperature superposition (TTS) principle. This principle is based on numerous experimental data and states that with the change in temperature flie relaxation spectrum as a whole shifts in a self-similar manner along t axis. Therefore, dynamie functions corresponding to different temperatures are similar to each otiier in shape but are shifted along the frequency axis by the value a flie latter is named the temperature-shift factor. With war for an argument it becomes possible to plot temperature-invariant curves Re G (War) and lm G, (war). The temperature dependence of a is defined by the formula... 

The splitting of these separatrix surfaces b studied in [164], [165. The behaviour of the solutions of a perturbed problem was studied using computer in the interesting paper [166]. In the schemes obtained in the calculations and showing the behaviour of integral trajectories it b clearly seen that the invariant curves of the unperturbed problem become chaotic in the neighbourhood of separatrices. Thb... 

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