Linked limit cycles

The limit bundle shown in the abstract model (1976-2) is also generically a multi periodic limit cycle. In the abstract model (1974-4) linked limit cycles are also observed. These linked limit cycles are examples of the generic multi limit cycle shown in Fig. IV.4. 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

So far we have concentrated on the particular parameter values <7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. 

It was pointed out earlier that oscillations in NDR oscillators are linked to three features of the electrochemical system (1) an N-shaped steady-state polarization curve (2) a resistance in series with the working electrode, which must not be too large and (3) a slow recovery of the electroactive species, in most cases due to slow mass transport. Hence, for every system that was discussed in the context of the possible origin of N-shaped characteristics, conditions can be estabhshed under which stable limit cycles exist, and for most of the systems mentioned, oscillations were in fact observed. This unifying approach was first put forth by Koper and Sluyters, and numerous experimental examples of electrochemical oscillations that can be deduced according to this mechanism are discussed in Ref. 60. 

All results established in chapter 2 for the analysis of eqns (2.7) in the phase plane therefore also apply to the two-variable system (5.4b,c). In particular, the two nullclines of the system remain defined by eqns (2.21). When parameter A exceeds a critical value, the product nullcline y= 4> takes the form of a sigmoid possessing a region in which the slope (da/dy) is negative in the phase plane (a, y). When the substrate nullcline v = a intersects the product nullcline in that region, in such a manner that condition (2.26) is satisfied, the steady state, which lies at the intersection of the two nullclines, is unstable and the system evolves towards a limit cycle corresponding to sustained oscillations (fig. 2.13). Such behaviour translates here into oscillations of ATP and extracellular cAMP, to which are associated oscillations in intracellular cAMP because of relation (5.4a), which links to the slower variables a and y. 

An example of a rhythm for which the existence of a limit cycle can still be questioned is the process of ovulation. A theoretical model shows (Lacker, 1981 Lacker, Feuer Akin, 1989) how the follicular phase of the menstrual cycle can be viewed in dynamic terms as the selection of an oocyte leading, in about two weeks, to ovulation. The latter, discontinuous process is followed by the degeneration of the luteal body, which corresponds to the luteal phase of the cycle. After the end of that phase, a new follicular phase begins. The periodicity of 28 days of the menstrual cycle in women thus follows from the succession of the follicular and luteal phases, each of which lasts 14 days. Separating these phases is the brief discontinuity of ovulation. The absence of ovulation is necessarily linked to the arrest of the hormonal rhythms observed in the course of a normal cycle. 

Thus we see how certain biological rhythms can be viewed as a cyclic succession of discontinuously linked events rather than as continuous oscillations of the limit cycle type. As demonstrated by the biochemical... 

Approximate analytical relationship were developed to predict the period and the amplitude of the limit cycle in a wide range of operation conditions. These theoretical results were linked to experimental findings. 

In spite of the link which might be established between the limit cycle type behaviour and the occurrence of target patterns, it is tempting to search very quickly for some qualitative correlation between the two. One can hopefully imagine, for instance, that a difference in the properties of the limit cycle has something to do with the above-mentioned fact that oscillations in thin layer do not necessarily give rise to centres. 

In the case where the physical system can be adequately modeled by a dynamical system on the plane, precise mathematical meaning can be given to this feature of physical robustness , and this was done by Andronov. First of all, he applied the Poincare theory of limit cycles and the Lyapunov theory of stability for studying modeling equations that allowed him and Vitt to explain many real phenomena in radio-engineering. Then, he linked the... 

In summary, therefore, solution and fiber biochemistry have provided some idea about how ATP is used by actomyosin to generate force. Currently, it seems most likely that phosphate release, and also an isomerization between two AM.ADP.Pj states, are closely linked to force generation in muscle. ATP binds rapidly to actomyosin (A.M.) and is subsequently rapidly hydrolyzed by myosin/actomyosin. There is also a rapid equilibrium between M. ADP.Pj and A.M.ADP.Pj (this can also be seen in fibers from mechanical measurements at low ionic strength). The rate limiting step in the ATPase cycle is therefore likely to be release of Pj from A.M.ADP.Pj, in fibers as well as in solution, and this supports the idea that phosphate release is associated with force generation in muscle. 

Selected entries from Methods in Enzymology [vol, page(s)] Analysis of GTP-binding/GTPase cycle of G protein, 237, 411-412 applications, 240, 216-217, 247 246, 301-302 [diffusion rates, 246, 303 distance of closest approach, 246, 303 DNA (Holliday junctions, 246, 325-326 hybridization, 246, 324 structure, 246, 322-324) dye development, 246, 303, 328 reaction kinetics, 246, 18, 302-303, 322] computer programs for testing, 240, 243-247 conformational distribution determination, 240, 247-253 decay evaluation [donor fluorescence decay, 240, 230-234, 249-250, 252 exponential approximation of exact theoretical decay, 240, 222-229 linked systems, 240, 234-237, 249-253 randomly distributed fluorophores, 240, 237-243] diffusion coefficient determination, 240, 248, 250-251 diffusion-enhanced FRET, 246, 326-328 distance measurement [accuracy, 246, 330 effect of dye orientation, 246, 305, 312-313 limitations, 246,... 

Anthracyclines are antitumor quinone containing antibiotics produced by different strains of Streptomyces. Some of them, such as adriamycin doxorubicin), and daunorubicin are broad spectrum antitumor compounds. They act by binding to DNA and interfering with DNA replication and gene transcription. Their limitations for clinical use are cardiac toxicity and drug resistance phenomena. Consequently, intense structure-activity relationship studies have been performed to improve the pharmacological profile as well as to enhance the affinity for DNA. In particular, a number of fluorinated anthracyclines have been prepared with introduction of fluorine atoms into D or A cycles, and into the aglycone side chain linked atC-14. ... 

The depressive phase of manic-depressive disorder often requires concurrent use of an antidepressant drug (see Chapter 30). Tricyclic antidepressant agents have been linked to precipitation of mania, with more rapid cycling of mood swings, although most patients do not show this effect. Selective serotonin reuptake inhibitors are less likely to induce mania but may have limited efficacy. Bupropion has shown some promise but—like tricyclic antidepressants—may induce mania at higher doses. As shown in recent controlled trials, the anticonvulsant lamotrigine is effective for many patients with bipolar depression. For some patients, however, one of the older monoamine oxidase inhibitors may be the antidepressant of choice. Quetiapine and the combination of olanzapine and fluoxetine has been approved for use in bipolar depression. 

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