Attractor reconstructing

These same issues confronted scientists in the mid-1970s. At the time, the only known examples of strange attractors were the Lorenz attractor (1963) and some mathematical constructions of Smale (1967). Thus there was a need for other concrete examples, preferably as transparent as possible. These were supplied by Henon (1976) and Rdssler (1976), using the intuitive concepts of stretching and folding. These topics are discussed in Sections 12.1-12.3. The chapter concludes with experimental examples of strange attractors from chemistry and mechanics. In addition to their inherent interest, these examples illustrate the techniques of attractor reconstruction and Poincare sections, two standard methods for analyzing experimental data from chaotic systems. 

Roux et al. (1983) exploited a surprising data-analysis technique, now known as attractor reconstruction (Packard et al. 1980, Takens 1981), The claim is that for systems governed by an attractor, the dynamics in the full phase space can be reconstructed from measurements of just a single time series Somehow that single variable carries sufficient information about all the others. 

The key to the analysis of Roux et al. (1983) is the attractor reconstruction. There are at least two issues to worry about when implementing the method. 

Many people find it mysterious that information about the attractor can be extracted from a single time series. Even Ed Lorenz is impressed by the method. When my dynamics class asked him to name the development in nonlinear dynamics that surprised him the most, he cited attractor reconstruction. 

In principle, attractor reconstruction can distinguish low-dimensional chaos from noise as we increase the embedding dimension, the computed correlation dimension levels off for chaos, but keeps increasing for noise (see Eckmann and Ruelle (1985) for examples). Armed with this technique, many optimists have asked questions like. Is there any evidence for deterministic chaos in stock market prices, brain waves, heart rhythms, or sunspots If so, there may be simple laws waiting to be discovered (and in the case of the stock market, fortunes to be made). Beware Much of this research is dubious. For a sensible discussion, along with a state-of-the-art method for distinguishing chaos from noise, see Kaplan and Glass (1993). 

Numerically integrate the Rossler system for a = 0.4, 6=2, c = 4, and obtain a long time series for x(f). Then use the attractor-reconstruction method for various values of the delay and plot (x(f), x(f + t)). Find a value of t for which the reconstructed attractor looks similar to the actual Rossler attractor. How does that T compare to typical orbital periods of the system ... 

Figure C3.6.1 (a) WR single-handed chaotic attractor for k = 0.072. This attractor is projected onto the (Cj,C2) plane. The maximum value reached by Cj(t) is ( 54.1 and the minimum reached by tT 2.5. The vertical line, at Cj = 8.5 for dj < 1, shows the position of the Poincare section of the attractor used later, (b) A projection, onto the plane, of the chaotic attractor reconstructed from the set of delayed coordinates c ft),c ft, c ...

Figure Cl.b.lCal shows an experimental chaotic attractor reconstructed from the Br electrode potential, i.e. the logarithm of the Br ion concentration, in the BZ reaction [17]. Such reconstruction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements X (t) / = 1,...

Figure 5 shows the 3-dimensional reconstructed attractors and their projections on canonical planes. The reconstructed phase portraits do not exhibit a defined structure, i.e., it is not toroidal or periodic. As matter of fact, the oscillatory structure is only observed in the Poincare map. The Poincare map is often used to observe the oscillatory structure in dynamical systems. The... 

Fig. 5. Reconstructed space phase (zi,Z2,Z3) from time series in Figure 3 and 4. The attractors seem ordered in layers (see projections in canonical planes). The attractor for concentrate vinasses (experiment E2.b, see text for details) is smallest. zi,Z2, Z3 are dimensionless.

A set of experiments on gas-liquid motion in a vertical column has been carried out to study its d3mamical behavior. Fluctuations volume fraction of the fluid were indirectly measured as time series. Similar techniques that previous section were used to study the system. Time-delay coordinates were used to reconstruct the underl3ung attractor. The characterization of such attractor was carried out via Lyapunov exponents, Poincare map and spectral analysis. The d3mamical behavior of gas-liquid bubbling flow was interpreted in terms of the interactions between bubbles. An important difference between this study case and former is that gas-liquid column is controlled in open-loop by manipulating the superficial velocity. The gas-liquid has been traditionally studied in the chaos (turbulence) context [24]. 

Figure 3.5 The Rossler strange attractor. (A) The phase space. (B) The state variable yi (t). (C) Reconstruction in the pseudophase space.

Although the detailed features of the interactions involved in cortisol secretion are still unknown, some observations indicate that the irregular behavior of cortisol levels originates from the underlying dynamics of the hypothalamic-pituitary-adrenal process. Indeed, Ilias et al. [514], using time series analysis, have shown that the reconstructed phase space of cortisol concentrations of healthy individuals has an attractor of fractal dimension dj = 2.65 0.03. This value indicates that at least three state variables control cortisol secretion [515]. A nonlinear model of cortisol secretion with three state variables that takes into account the simultaneous changes of adrenocorticotropic hormone and corticotropin-releasing hormone has been proposed [516]. 

Fig. 5.36. Reconstructed attractors corresponding to time series in Fig. 5.35. (Reprinted with permission from references [66 and 67], Royal Society of Chemistry and American...

Note that in each case the method gives a closed curve, which is a topologically faithful reconstruction of the system s underlying attractor (a limit cycle). 

For this system the optimum delay is t = f, i.e., one-quarter of the natural orbital period, since the reconstructed attractor is then as open as possible. Narrower cigar-shaped attractors would be more easily blurred by noise. ... 

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