Quasiperiodic regime

Figure 2. Schematic of typical data and consistent Poincare sections from the quasiperiodic regime of Rayleigh-B nard convection. The rotation number W (in arbitrary units) is plotted versus Rayleigh number R for two different values...

More testing of these approaches is certainly required before any final pronouncements on their worth can be made. One important contribution to assessing the ideas was that of Sewell et al. (53). For a system of coupled oscillators, they showed that the constraints of Miller et al. and Bowman et al. (45,46) may induce physically undesirable effects. In particular, they gave an example of a trajectory in the quasiperiodic regime being transformed into a chaotic one by the action of the constraints. 

Chaos may also occur as a consequence of the destruction of a two-torus that characterizes a quasiperiodic regime with two incommensurate frequencies. Quasiperiodicity in chemistry was recently discovered in experiments on the BZ reaction, and chaos was reached through the development of wrinkles on the torus (ROUX and ROSSI, this volume [52]). ARGOUL et al, [53] (in this volume) have proposed a tentative interpretation of the experiments. 

Inside the resonant wedge all trajectories on the invariant torus tend to a stable periodic orbit, which means that the dominating regimes here is a periodic one. Outside the wedge either a quasiperiodic regime or a periodic one of a very long period is established on the torus. Both are practically indistinguishable. Therefore, a transition over the boundary of a resonant... 

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

We remark that an expression like (4.13) can no longer be derived in general because of the interference between the two amplitude terms in (4.16). This is the general feature of a nonseparable regime where the spectrum of resonances loses its regularity. When there exist two fundamental periodic orbits, we may expect that the spectrum of resonances still displays quasiperiodic regularities, as is the case for the three-disk scatterer [33]. 

Figure 8.3. Quasiperiodic oscillation regime for the reaction intermediates according to the Oregonator kinetic model.

Thus, it appeared naturally to assume that every interesting dynamical regime possesses a discrete frequency spectrum. In this connection, it is curious to note that Landau and Hopf had proposed quasiperiodic motions with a sufficiently large number of independent frequencies as the mathematical image of hydrodynamical turbulence (the number of the frequencies was supposed to increase to infinity as some structural parameter, such as the Reynolds number, increases). 

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