Complex time series

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). 

It may appear that Table 1 contains an essentially complete summary of patterns that may form in electrochemical systems. This impression is misleading, since Table 1 only roughly summarizes results observed so far or predicted with models. These are investigations concentrating on phenomena that can be described with two essential variables (two-component systems). This survey is certainly not yet completed. Furthermore, numerous examples of current or potential oscillations involve complex time series. Only in a few cases does the complex time series result from the spatial patterns. In most cases, the additional degree of freedom will be from a third dependent variable, such as from a concentration that adds an additional feedback loop into the system, as discussed in Section 3.1.3. Spatial pattern formation in three-variable systems is an area that currently develops strongly in nonlinear dynamics. In the electrochemical context, the problem of pattern formation in three-variable systems has not yet been approached. 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

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. 

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. 

Non-linear steady state bistability in Economics and Physiology Oscillations spatio-temporal oscillatory features in population Chaos and highly complex time series in Sociology and Economics Fractal growth of cities... 

One of the most common ways of characterizing complexity is by taking Fourier transforms. The spatial power spectrum of a time series of [Pg.394]

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

The first level of complexity corresponds to simple, low uncertainty systems, where the issue to be solved has limited scope. Single perspective and simple models would be sufficient to warrant with satisfactory descriptions of the system. Regarding water scarcity, this level corresponds, for example, to the description of precipitation using a time-series analysis or a numerical mathematical model to analyze water consumption evolution. In these cases, the information arising from the analysis may be used for more wide-reaching purposes beyond the scope of the particular researcher. 

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... 

The nitrate time series of the drinking water reservoir is very complex and demonstrates the difficulty of identifying such models. A model with the shape ARIMA (p,d,f) (sp,sd,sf) shall be estimated. 

At the same time, it is necessary to take into account that the approach described has a number of exceptions, related for example to the nature of other ligands forming pseudohalide complexes. A series of classic examples of inversion of the bond M — N —> M — S —> M — N have been reported [6,8,11,42-44,59] and are presented in Sec. 2.2.3.5. In this respect, we especially emphasize the capacity of other ligands for soft or hard metals, related with symbiotic [60] and anti-symbiotic [61] effects. Thus, Pearson [61] emphasized that soft ligands, which are placed in a trans position to SCN ion, contribute to N-binding of thiocyanate ions, and hard bases contribute to S-coordination of these ambidentate ligands. Metal oxidation number (Table 1.4) is important in this problem and it regulates soft hard properties of complex-formers. 

To illustrate the occurrence of intra-nephron synchronization in our experimental results, Fig. 12.13 shows that ratio fsiow/fjast as calculated from the time series in Fig. 12.2c. We still observe a modulation of the fast mode by the slow mode. However, the ratio of the two frequencies maintains a constant value of approximately 1 5 during the entire observation period, i.e., there is no drift of one frequency relative to the other. In full agreement with the predictions of our model, data for other normotensive rats show 1 4 or 1 6 synchronization. Transitions between different states of synchronization obviously represent a major source of complexity in the dynamics of the system. It is possible, for instance, that a nephron can display ei-... 

In real spectra, there will be several peaks, and the time series appear much more complex than in Figure 3.16, consisting of several superimposed curves, as exemplified in Figure 3.17. The beauty of Fourier transform spectroscopy is that all the peaks can... 

The process of Fourier transformation converts the raw data (e.g. a time series) to two frequency domain spectra, one which is called a real spectrum and die other imaginary (diis terminology conies from complex numbers). The true spectrum is represented only by half the transformed data as indicated in Figure 3.18. Hence if there are 1000 datapoints in the original time series, 500 will correspond to the real transform and 500 to the imaginary transform. 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

Liquids and proteins are complex systems for which the smdy of dynamical systems has wide applicability. In the conference, relaxation in liquids (s-entropy by Douglas at the National Institute of Standards and Technology, nonlinear optics by Saito, and energy bottlenecks by Shudo and Saito), energy redistribution in proteins (Leitner and Straub et al.), structural changes in proteins (Kidera at Yokohama City University), and a new formulation of the Nose-Hoover chain (Ezra at Cornell University) were discussed. Kidera s talk discussed time series analyses in molecular dynamics, and it is closely related to the problem of data mining. In the second part of the volume, we collect the contributions by Leitner and by Straub s group, and the one by Shudo and Saito in the third part. 

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