Existence of a fixed point

When the process has been iterated long enough (m large), the subunits ate large, and we know from the properties of dilute solutions that two 

The sequence of numbers approaches a fixed point u when 

In most practical cases this fixed point is reached quickly. For example, widi g = 3 and m = 4 we are already dealing with subunits of 3 100 monomers, for which the hard sphere limit usually holds well. 

The fixed-point theorem [eq, (XI.11)] is essential because when ii. reaches its limiting value u, the relationship between a , and Om-t [eq. (XI.9)] becomes a simple geometric series 

16) we prove the existence of an exponent v. Furthermore, we have an expiicit value for v, derived from eq. (XI. IS) 

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For contractive functions the existence of a fixed point can be shown ... 

So, we can now simply apply the results of the previous section. Thus, if the family (11.3.4) is in a general position (i.e. the rank of the matrix (11.2.18) is maximal if I2 0, this condition reduces to the inequality (11.2.9)), then the set of parameter values which corresponds to the existence of a fixed point with a imit multiplier and zero Lyapunov values Z2> , Zjb i, forms a smooth surface of codimension (fc — 1) that passes through e = 0, The families of maps transverse to SDt can be recast into the form... 

The conditions for the existence of a pT-periodic solution to the forced system can be written in terms of a fixed point equation for the pth iterate of the stroboscopic map... 

One immediately sees a major difference with the flow equation Eq. (31) for d = 1. Because of the change of sign of the quadratic term in Eq. (45), there is now a fixed point for d = 1 as shown in Fig. 6(b). This flow equation does not behave properly in a range d 6 [1.5,2) but that is more of a problem of implementation of RG than directed polymer per se, and so, may be ignored here. Note also that no extra information can be obtained from Eq. (45) for d > 2 other than what we have obtained so far in Sec. 4, namely the existence of a critical point. 

In Section 1.4 the topological concept of rotation was used to prove the existence of equilibrium states. When the reaction kinetics are not restricted by Postulate 1.5.1, each invariant manifold may include more than one equilibrium states and it is interesting to obtain information about the number and stability of these states. In the present section we shall use one more topological concept, the index of a fixed point, to show that the equilibrium states are odd in number, 2m+1, among which m at least are unstable. As in the preceding sections, the discussion concerns isolated systems, but extension to other closed systems should not present difficulties. 

Theorem A.l [23]. Let be a completely continuous vector field having a finite number of null points in the interior of a region V and no null points on the boundary dV. Then the rotation of on 0Kis equal to the sum of the indices of the null points. In particular, if the rotation differs from zero, there exists at least one null point of (a fixed point of H) in V. 

Theorem 1.21. If S is a contraction mapping in a Hilbert space V then there exists a fixed point u such that Su = u and solutions u of the equation... 

The theorem of existence is proved by finding a fixed point of the following operator (which is not compact, in general). Taking = TL... 

The KTTS depends upon an absolute 2ero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equiUbrium, together with specification of an interpolation instmment and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

A consequence of single-ion diffusion is that the mass movement must be compensated for by an opposing drift (relative to a fixed point deep in the metal) of the existing oxide layer if oxidation is not to be stifled by lack of one of the reactants. The effect may be illustrated by reference to a metal surface of infinite extent (Fig. 1.81). 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

The previous discussion has shown us how to calculate the total number of possible cyclic states. We also know, from Lemma 2, that all cycle lengths must divide the maximal cycle length Hiv obtain the exact number of distinct cycles and their lengths takes a little bit more work. If flw prime, we know that the only possible cyclic lengths are 1 and It can then be shown that only the null configuration is a fixed point unless N is some multiple of 3, it which case there are exactly four distinct cycles of length one. If Hat i ot prime, there can exist as many cycles as there are divisors of Although there is no currently known closed form... 

Let be a Kahler manifold with a holomorphic symplectic form cjc- Suppose there exists a C -action on X with the property that tplujc = tuJc for t G C, where we denote the C -action on X hy il)t X X. Let C, be a connected component of the fixed point set of the C -action, and consider the subset defined by... 

Proof. Let I E be a fixed point of the C -action. There exist distinct points... 

Now let us consider the fixed point set of the C -action. In the case of A/"s, the obvious component of the fixed point set is Afs which corresponds to the set of a point of the form ( /,0) G A/s- There exist other components, and this shows that A/s is not exactly the same as T Afs- These components are described as follows. By definition E, ) G A/s is a fixed point if and only if there exists an isomorphism between ( , ) and E,t ). This... 

So, the phase space V is divided onto subsets Att(Cy). Each of these subsets includes one cycle (or a fixed point, that is a cycle of length 1). Sets Att(Cy) are (jf)-invariant (jf)(Att(Cy)) C Att(Cy). The set Att(Cy) Cy consist of transient points and there exists such positive integer t that (jf) (Att(Cy)) — Cj if... 

The decomposition of turbulent motion into mean and random fluctuations resulting in the separation of the flux, Eq. 22-27, leaves us a serious problem of ambiguity. It concerns the question of how to choose the averaging interval s introduced in Eq. 22-24. In a schematic manner we can visualize turbulence to consist of eddies of different sizes. Their velocities overlap to yield the turbulent velocity field. When these eddies are passing a fixed point, they cause fluctuation in the local velocity. We expect that some relationship should exist between the spatial dimension of those eddies and the typical frequencies of velocity fluctuations produced by them. Small eddies would be connected to high frequencies and large eddies to low frequencies. 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

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