Quasiperiodic behavior

Quasiperiodic behavior, Pt, 37 240-242 Quasi-stationary concentrations, reaction mechanisms by intermediates in, 5 311... 

They reduce the set (5) of four equations in real variables to two equations. This means that we can have only regular, periodic, or quasiperiodic behavior, never chaos. Chaos in a dynamical system governed by ordinary differential equations can arise only if the number of equations is equal to or greater than 3. We remember that we refer to the case of perfect phase matching (Afe = k — 2fe2 = 0), and the well-known monotonic evolution of fundamental and... 

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. 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

As a summary for the second study case, the bubble streams interactions induce the presence of complex oscillatory phenomenon [51]. For instance, since the bubble rise almost-linearly into the plume, the presence of the central plume induce an almost-periodic behavior. We shown that the number of fundamental frequencies (which are in some sense induced by the modes of the bubble streams) increases when the superficial gas velocity increases yielding a route to chaos (periodic- quasiperiodic-chaotic behavior). 

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. 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. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as t —> . For practical reasons, we should require that such trajectories are not too rare. For instance, we could insist that there be an open set of initial conditions leading to aperiodic trajectories, or perhaps that such trajectories should occur with nonzero probability, given a random initial condition. 

Quasiperiodicity is significant because it is a new type of long-term behavior. Unlike the earlier entries (fixed point, closed orbit, homoclinic and heteroclinic orbits and cycles), quasiperiodicity occurs only on the torus. 

It is clear, from the discussion thus far, that typical molecular Hamiltonians display features characteristic of chaotic classical motion. The logical order followed previously, that is, the introduction of well-defined concepts of chaotic behavior in classical ideal systems, followed by an examination of realistic molecular models, does not follow through to quantum mechanics. The primary difficulty is that quantum mechanics always predicts, for bound-state dynamics, quasiperiodic motion. Several aspects of quantum chaos are discussed in Section IV. We note at this point, however, that this quantum-... 

Thus, after a transient time the spatial pattern becomes time-periodic (or quasiperiodic when pi is complex) with an exponentially decaying amplitude. Such surprisingly simple periodic behavior in time, in sharp contrast with the complex irregular spatial structure of the concentration field, has been first observed in numerical simulations by Pierrehumbert (1994) and later was also confirmed experimentally by Rothstein et al. (1999). In analogy with the rich multiscale spatial... 

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