Chaotic time series

Chaotic time series have been obtained from a wide variety of experimental systems. Figures 16 and 17 show examples of the irregular time series found in various cases. It is somewhat problematic to assign the term chaotic to these reported time series because it was rarely investigated whether the time series were deterministically chaotic in the strict sense of the word. Therefore, when we use the expression chaotic, we are well aware that there is, in many cases, no proof for chaos in the oscillation patterns. The few reports wherein a thorough analysis has been performed (64,104) do though show the existence of deterministically chaotic oscillations. It would be beyond the scope of this review to describe the methods... 

Fig. 18. Evolution of chaotic time series with decreasing oxygen concentration (from a to i) during ammonia oxidation over a Pt wire. (From Ref. 211.)...

Lorenz s idea i s that z should predict. To check this, he numerically integrated the equations for a long time, then measured the local maxima of z(t), and finally plotted z . vs. z As shown in Figure 9.4.3, the data from the chaotic time series appear to fall neatly on a curve—-there is almost no thickness to the graph ... 

Nonlinear Analyses of Periodic and Chaotic Time Series from die Pmxidase— Oxidase Reaction. 

The power spectrum is probably the most frequently used measure. The power spectmm decomposes a complex time series or system behavior into a series of sine waves of a range of frequency. The power spectrum of a chaotic time series is normally a continuous power spectrum. The power associated with a particular frequency is the square of the amplitude of that sine function. A time series is chaotic with a broad band of continuous frequency with appreciable power. 

In the case of complex periodic and chaotic time series, role of seasonal variation and random variation can be quite important. 

The Mackey-Glass Time Series is a chaotic time series proposed by Mackey and Glass [25]. It is obtained from this non-linear equation ... 

Figure A3.14.7. Example oscillatory time series for CO + O2 reaction in a flow reactor corresponding to different P-T locations in figure A3,14,6 (a) period-1 (b) period-2 (c) period-4 (d) aperiodic (chaotic) trace (e) period-5 (1) period-3.

Figure C3.6.4(a) shows an experimental chaotic attractor reconstmcted from tire Br electrode potential, i.e. tire logaritlim of tire Br ion concentration, in tlie BZ reaction [F7]. Such reconstmction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements (A (tj) / = 1,...

In order to analyze both systems, some techniques from nonlinear science are burrowed. Firstly, a phase portrait is constructed from delay coordinates, a Poincare map is also computed, FFT is exploited to derive a Power Spectrum Density (PSD) Maximum Lyapunov Exponents (MLE) are also calculated from time series. Although we cannot claim chaos, the evidence in this chapter shows the possible chaotic behavior but, mostly important, it exhibits that the oscillatory behavior is intrinsically linked to the controlled systems. The procedures are briefly described before discuss each study case. 

Figure 6 displays the Poincare maps for all experiments. Note that even the projections in canonical planes (see Figure 5) seem ordered in layers. That is, a toroidal structure can be seen form the Poincare surface. That is, small amplitude oscillations were detected in time series (see Figures 3 and 4) for all experiments. The t3rpical behavior of aperiodic (possibly chaotic) oscillations can be confirmed is one takes a look at the corresponding Poincare section... 

Essentially, MLE is a measure on time-evolution of the distance between orbits in an attractor. When the dynamics are chaotic, a positive MLE occurs which quantifies the rate of separation of neighboring (initial) states and give the period of time where predictions are possible. Due to the uncertain nature of experimental data, positive MLE is not sufficient to conclude the existence of chaotic behavior in experimental systems. However, it can be seen as a good evidence. In [50] an algorithm to compute the MLE form time series was proposed. Many authors have made improvements to the Wolf et al. s algorithm (see for instance [38]). However, in this work we use the original algorithm to compute the MLE values. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

In Fig. 6 the Br time series for one sequence of alternating states are shown (Fig. 6a-c) together with the corresponding power spectra (Fig. 6d-f). Figure 6a illustrates the first complex periodic state (one RO, one QHO) which appears as t is increased from 0.294 hr. The second complex periodic state (one RO, two QHO) is shown in Fig. 6c, and the intervening chaotic state in Fig. 6b. Each periodic state is characterized by a power... 

Figure 35 shows a typical example for a time series in which slight variation of one of the external parameters caused a transition from regular harmonic behavior to period doubling (147). Besides even more complex time series, irregular oscillations suggesting chaotic behavior were also observed, but have not yet, however, been fully analyzed as has Pt(l 10). 

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 ... 

Nearly all models discussed so far have one feature in common they are not distributed models and can describe only spatially uniform systems. Many of the mathematical models use ordinary differential equations, and the resultant time series are nearly always simply periodic. This approach, however, describes only part of the experimentally observed behavior there is a great deal of experimental evidence for spatial heterogeneity and chaotic oscillatory behavior in heterogeneously catalyzed systems. 

Complex time series have also been detected for Pd 129,131). In these studies, however, it was shown that the complex structure could be characterized by three or fewer superimposed frequencies (729) and was thus not deterministically chaotic. Chaotic oscillations have been reported for several other sterns, namely, NH3/O2 on Pt (40,211,212), H2/O2 on Ni (168,169) and Pt (752), CO/NO on Pt and Pd (91,123), C2H4/H2 on Pt (335), C3H6/O2 on Pt (194), l-hexene/02 on Pd (798), and the CH3NH2 decomposition over Pt, Rh, and Ir (24). This list, while incomplete, shows that irregular oscillations and often chaos are frequently encountered during heterogeneous catalytic studies. 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

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