Such a critical fixed point is called a complex degenerate) saddle. Its stable manifold is y = 0, and the unstable manifold is given by x = 0, as shown in Fig. 10.2.6(b). Here, in the critical case, the trajectories behave qualitatively identical to those nearby the rough unstable cycle shown in Fig. 10.2.7(b). [Pg.117]

As shown in Chap. 5, the critical fixed point 0(0,0) lies in an invariant C" -smooth center manifold defined by the equation y = (x), where vanishes at the origin along with its first derivative. Moreover, the following reduction theorem holds [Pg.110]

For critical process control parameters specified by fixed points, the value ought to be challenged within an acceptable tolerance, typically 1 unit. [Pg.825]

Infinite-Randomness Quantum Ising Critical Fixed Points. [Pg.216]

The basic tool for studying stability of critical fixed points is the Lyapunov functions. [Pg.111]

Up to now we tacitly assumed that the single fixed point t> dominates the critical manifold. This is not the full story. In the applications of interest there are at least two fixed points located on the critical manifold. Besides t)J there is a fixed point vj at which an additional coordinate [Pg.172]

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. [Pg.179]

Fig. 10.2. Schematic flow diagram on the critical surface close to a fixed point. Full lines show the flow of nonlinear scaling fields, where > A0 > 0. That relation is signalled by the broken lines which illustrate that the flow from a generic point on the critical surface approaches the fixed point from direction. The tangents of |

Temperature measurement(s), 24 433-467, 75 469, 77 783-784 of critical current density, 23 847-848 fixed-point thermometer calibration, [Pg.926]

If we choose a generic element we have exp R = T. Therefore in such a case, the critical point is the same as the fixed point of the torus action. In the following, we assume is generic, and hence Crit(/) = [Pg.52]

We have seen in Sec. 10.4 that in the case of weak resonance cj = 2nM/N N > the stability of the critical fixed point is, in general, determined by the sign of the first non-zero Lyapunov value. The same situation applies to the critical case of an equilibrium state with a purely imaginary pair of characteristic exponents. However, there is an essential distinction, namely, for a resonant fixed point only a finite number which does not exceed N—3)/2 of the Lyapunov values is defined. The question of the structure of a small neighborhood of the fixed point in the case where all Lyapunov values vanish is difficult, so we do not study it here. Instead, we consider two examples. [Pg.159]

The critical behavior is, however, the same there is a Kosterlitz-Thouless (KT) transition at the phase boundary Ku between a disorder dominated, pinned and a free, unpinned phase which terminates in the fixed point K = 6/p2. One can derive an implicit equation for Ku by combining (23a) and (23b) to a differential equation [Pg.101]

It is evident that the projections of the fixed points A and B, in the EJ map, always lie on relative equilibria of type I, but that the position of the overlapping projections of C and D depends on the sign of b — a. If > b the double point is isolated between the two type I equilibria, and quantum monodromy is expected, for a sufficiently dense quantum lattice. If, on the other hand, the critical point lies on the type II relative equilibrium line and [Pg.74]

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. [Pg.218]

We are now in the position to sketch the RG flow globally. A schematic picture of the flow in the critical manifold is shown in Fig. 10.2, where the manifold is approximated as a plane parameterized by wfy, wfy J. Assuming that is the irrelevant field of smallest fixed point dimension we note that [Pg.170]

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p- [Pg.356]

As 9 continues to increase (1 +< 9), the characteristic multipliers 2C an(l 2d Pa-ss through = — 1, and so these 2-period solutions become unstable. At this stage, we look at the fourth iterate and we find, as might now be expected, that a 4-cycle periodic solution appears (Figure 3.3 C, fixed point of period 4 for 9 = 3.5). The period doubles repeatedly and goes to infinity as one approaches a critical point 9C at which instability sets in for all periodic solutions, e.g., for the model (3.1), 9C 3.5699456. Above 9C all fixed points are unstable and the system is chaotic. [Pg.49]

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