Fixed point behavior

Langton wais able to provide a tentative answer to his question by examining the behavior of the entire rule space of elementary one-dimensional cellular automata rules (see discussion in section 3.2) as parameterized by a single parameter A. He found that as A is increased from its minimal to maximal values, a path is effectively traced in the rule space that progresses from fixed point behavior to simple period-... 

Since —. 472 we conclude that the leading corrections to fixed point behavior in the strong coupling regime as described by our theory are proportional to and thus dominate the nonuniversal corrections... 

Many, possibly all, rules appear to generate asymptotic states which are block-related to configurations evolving according to one of only a small subset of the set of all rules, members of which are left invariant under all block transformations. That is, the infinite time behavior appears to be determined by evolution towards fixed point rule behavior, and the statistical properties of all CA rules can then, in principle, be determined directly from the appropriate block transformations necessary to reach a particular fixed point rule. 

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-... 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

Figures 2 and 3 show typical test results for flux decline in laminar flow where the pressure and temperature are varied and the Reynolds number is held fixed. Similar behaviors are found with variations in Reynolds number and for turbulent flow. The important feature of the data is that the flux decline is exponential with time and an asymptotic equilibrium value is reached. Each solid curve drawn through the experimental points is a least-square fit exponential curve defined by Eq. (19). It is interesting to note that Merten et al ( ) in 1966 had observed an exponential flux decay in their reverse osmosis experiments. However, Thomas and his co-workers in their later experiments reported an algebraic flux decay with time (4,5).

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

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... 

Summary. We suggest a simple system of two electron droplets which should display two-channel Kondo behavior at experimentally-accessible temperatures. Stabilization of the two-channel Kondo fixed point requires fine control of the electrochemical potential in each droplet, which can be achieved by adjusting voltages on nearby gate electrodes. We study the conditions for obtaining this type of two-channel Kondo behavior, discuss the experimentally-observable consequences, and explore the generalization to the multi-channel Kondo case.1... 

Finally, while the channel asymmetry parameter is relevant in the RG sense, for realistically well-matched channel couplings we expect that the system will remain near the 2CK fixed point, and will show NFL behavior, over a wide range of temperatures. 

For larger T (T = 1.6), chaotic behavior arises, the hyperbolic fixed point is disrupted and the tori are perturbed (see Figure 1.20) [28]. A chaotic region appears with homoclinic tangle and formation of new hyperbolic and elliptic points. 

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

The asymptotic behavior of a two-dimensional autonomous dynamical system is, in general, known to have fixed points or lines where f = s = 0, or to have limit cycles where f, i 0. Since the system is irreversible as the concentrations of chiral products always increase at the cost of the substrate A ... 

The solutions of these equations (the trajectories) will for long times (i.e., after transient effects associated with switching on the external parameters have decayed) approach so-called limit sets, which may be classified into fixed points (stationary states), limit cycles (periodic oscillations), mixedmode oscillations, quasiperiodic oscillations, and chaotic behavior. Transitions between these states may occur upon variation of the external parameters pk and are called bifurcations. Experimental evidence for these effects with the system CO + 02/Pt(110) will be briefly presented without going further into details of the underlying general theory (see 16, 17). 

Fig. 13. Kinetic oscillations during the CO/O reaction on Pt(110) at I = 540 K, />0, = 7.5 x 10-5 torr, and for varying pm. (From Ref. 71.) (a) pco = 3.90 x 10 lorr constant behavior (fixed point), (b) pt0 = 3.K4 x I0"5 torr onset of harmonic oscillations with small amplitudes (Hopf bifurcation), (c)pco = 3.66 x 10 5 torr harmonic oscillation with increased amplitude, (d) pc0 = 3.61 x I0-5 torr first period doubling, (e) pc0 = 3.52 x 10 torr second period doubling, (f) pco = 3.42 x 10 5 torr aperiodic (chaotic) behavior.

---