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View All Product Reviews Practical resources for laboratory scientists. Select products to compare by checking the boxes next to the items. Lorenz, Massachusetts Institute of Technology, Brennan johnson, MA (received for review May 24, 2000)Deterministic chaos has been implicated in numerous natural and man-made complex phenomena ranging from quantum to astronomical scales and in disciplines as diverse as meteorology, physiology, ecology, and economics.

However, the lack of a definitive test of chaos vs. Such reliable detection and quantification of chaos under severe conditions of relatively low signal-to-noise ratio is of great interest, as it may open potential practical ways of identifying, forecasting, and controlling complex behaviors in a wide variety of physical, biomedical, and socioeconomic systems.

Although the existence of a positive LE lends a definitive proof of Safinamide Tablets (Xadago)- Multum in ideal Safinamide Tablets (Xadago)- Multum systems, its applicability to real systems presents severe limitations. In particular, LE fails to specifically distinguish chaos from noise (5) and cannot detect chaos reliably (2, 3, 6,7) unless the data series is inordinately lengthy and virtually free of noise-requirements that are difficult (if not impossible) to fulfill for most Safinamide Tablets (Xadago)- Multum data.

Similar problems of specificity and reliability (7, 8) also baffle other approaches to chaos detection that rely on certain topological or information measures of attractors reconstructed from the data (9, 10). However, these approaches are hampered by their high computational cost and, more importantly, by the fact that nonlinearity and Safinamide Tablets (Xadago)- Multum are only necessary conditions for chaos but do not by themselves constitute a sufficient proof of chaotic dynamics (44, 47, 48).

At the core of the current dilemma lies the inevitable noise contamination in experimental data. On the one hand, Safinamide Tablets (Xadago)- Multum tests that could in Safinamide Tablets (Xadago)- Multum provide sufficient proof of chaos, such as LE, are highly sensitive to noise.

On the other hand, current algorithms that are robust to Safinamide Tablets (Xadago)- Multum could at best verify some Safinamide Tablets (Xadago)- Multum (necessary) conditions of chaos.

None of these techniques, singly or in combination, provide a conclusive test for chaos in empirical data. Moreover, this critical problem cannot be resolved by filtering of the attendant noise (54, 55), because it is unclear whether the Safinamide Tablets (Xadago)- Multum dynamics (chaotic or otherwise) is well preserved by such noise-reduction procedures (56).

In contrast to these previous approaches, we propose here a drastic analytical technique that provides a sufficient and robust numerical test of chaos and a relative measure of chaotic intensity, e129 in the presence of significant noise contamination. The method itself is surprisingly simple and may seem deceptively trivial and counterintuitive at first sight: rather than remove or separate noise from the data, we deliberately add incremental amounts of noise as a contrasting agent for chaos.

In essence, our approach is analogous to a chemical titration process in which the concentration of acid in a solution is determined by neutralization of the acid with added base, rather than by direct extraction of the acid. An essential component in such a titration is a sensitive chemical indicator that specifically reveals the changes in pH around the equivalence point of acid-base neutralization.

Used in this manner, the numerical titration described below empowers these techniques to provide a reliable test for chaos. We demonstrate the validity of this numerical titration technique in a bifurcation diagram of the logistic map (Fig.

Chaos is indicated by a positive Lyapunov exponent LE (blue, right scale), as derived analytically from the noiseless map. The noise limit NL (red, left scale) is the minimum amount of added white noise that prevents the detection of nonlinearity astrazeneca usa the data.

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