Period-two cycle

Fig. 11.4.2. Transformations of the Lamerey spiral for the case l > 0. The unstable period-two cycle bounds the attraction basin of the origin.

Fig. 11.4.7. The map for the case h when moving counter-clock-wise in the direction around the origin in the bifurcation diagram in Fig. 11.4.6(a). Two period-two cycles in (c) coalesce on the border separating D2 a-nd Do and disappear in Do. The semi-stable cycle of period two is shown in (d).

Theorem 5 [goles87a] If the synaptic-weight matrix A is symmetric, and the number of sites in the lattice is finite, then the orbits of the generalized threshold rule (equation 5.121) are either fixed points or cycles of period two. 

Since theorem 5 applies to this system, we immediately conclude that majority can yield only either fixed points or cycles of period two. It turns out that one can actually prove a stronger result for this particular rule. Let a finite state be any state consisting of a finite number of nonzero sites. Then we have the following theorem. 

Equation 5.121 is then reproduced in full by setting bi =5-1/2 for all i. Since aij is symmetric, the condition of theorem 5 is met. We conclude that (f)2d majority can only yield either fixed points or cycles of period two. 

Although it is certainly not immediately obvious from the rule itself, it turns out that, just as is the case for generalized threshold rules, the only possible asymptotic states of finite symmetric muib -threshold rules are either fixed points or cycles of period two [golesQO]. Unlike their binary brethren, however, multi-threshold rules possess some intriguing additional properties. 

Metal triflate Lewis acids can also be dispersed in ionic liquids for catalytic applications. Acetylation of alcohols with acetic anhydride and acetic acid has been reported with Cu(OTf)2, Yb(OTf)3, Sc(OTf)3, In(OTf)3, HfClq. (THF)2, and InCl3 in ionic liquids that consist of [BMIM] and the anions BF4, PF, or SbF 166). With lmol% acid, all the catalysts in [BMIMJPF showed >99% acetylation products in acetyl anhydride acetylation of benzyl alcohol. Sc(OTf)3 showed the best yield with recycling, with a 25% drop in yield after two cycles. A relatively long reaction period was needed to obtain a high yield (95-98%) for the acetylation of benzyl alcohol with acetic acid, indicating that the activities of the catalysts were... 

This test is conducted to quickly obtain data indicating deterioration and is used instead of the vacuum stability test when the effect of polymers on the stability of highly sensitive explosives is to be studied. A 0.6 g explosive mixed with 0.6 g polymer and also explosive and polymer separately are heated for two cycles, each of 48-h period at 100 °C. A 5% loss during the two cycles, each of 48-h period, or an explosion within 100 hours indicates excessive deterioration. 

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

When the system is pulsationally unstable and in secular stabitli-ty, we can have the period two limit cycles and succeeding perioddoubling for the change of p. The values of the parameters are, for instance, as follows ... 

Cooling period Process cycles 30 days One Decontamination factors 250 days One 400 days Two ... 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

Our next-higher value of hf = 0.004552 exhibits a double-loop limit cycle, or a period-two periodic attractor in the phase plot of Figure 4.55. 

Fig. 12.10 Bifurcation diagram obtained from the model with the damping constant of arteriolar oscillation as a parameter. If the dampling is reduced, chaotic phenomena can arise at relatively low values of the TGF gain. T = 16 s and a = 24. For = 0.04 s 1 the model displays a period-4 cycle. The figure only follows two of the four branches of this cycle.

Fig. 16 are only rarely strictly periodic, because usually rather small fluctuations in the external parameters are sufficient to trigger abrupt changes. However, in principle, mixed-mode oscillations belong to the category of multiple-periodic limit cycles. If the behavior is governed by two incommensurate frequencies, i.e., the ratio of two periodicities is an irrational number. This situation is denoted by quasiperiodicity and has been realized experimentally with periodically forced oscillations, as will be described next. 

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