Chaos indicators

Lega, E. and Froeschle, C. (1997). Fast Lyapunov Indicators. Comparison with other chaos indicators. Application to two and four dimensional maps, in The Dynamical Behaviour of our planetary system., Kluwer academ. publ., J. Henrard and R.Dvorak eds. 

Our immediate and instinctive reaction to an impact or explosion leaves a mental image of utter chaos and destruction. There may be a fascination with the power of such events, but our limited time resolution and limited pressure-sensing abilities cannot provide direct information on the underlying orderly mechanical, physical, and chemical processes. As with other phenomena not subject to direct examination by our human senses, the scientific descriptions of shock and explosion phenomena rest upon a collection of images of the processes which are derived from a range of experiences. The three principal sources of these images in shock science—experiment, theory, and numerical simulation—are indicated in the cartoon of Fig. 3.1. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Lahey (1990) indicated the applications of fractal and chaos theory in the field of two-phase flow and heat transfer, especially during density wave oscillations in boiling flow. 

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

A. S. Verkman, M. C. Sellers, A. C. Chao, T. Leung, and R. Ketcham, Synthesis and characterization of improved chloride-sensitive fluorescent indicators for biological applications, Analyt. Biockem. 178, 355-361 (1989). 

Note CHAOSBASE does not prevent the introduction of substructures which are contradictory. It is possible, for example, to indicate that a nitrogen atom belongs to the functional group "Ketone". Nor does it prevent possible violations of valence. The consequences of defining an "impossible retron" are that the CHAOS program will never be able to detect its presence in the target molecule. 

In the atmosphere, the vapor pressure of the isomeric cresols, 0.11+0.30 mmHg at 25.5 °C (Chao et al. 1983 Daubert and Danner 1985), suggests that these compounds will exist predominantly in the vapor phase (Eisenreich et al. 1981). This is consistent with experimental studies that found all three isomers in the gas phase of urban air samples, but they were not present in the particulate samples collected at the same time (Cautreels and Vancauwenbergh 1978). The relatively high water solubility of the cresol isomers, 21,520- 25,950 ppm (Yalkowsky et al. 1987), indicates that wet deposition may remove them from the atmosphere. This is confirmed by the detection of cresols in rainwater (Section 5.4.2). The short atmospheric residence time expected for the cresols (Section 5.3.2.1) suggests that cresols will not be transported long distances from their initial point of release. 

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. 

Moreover, during part of the observation period (180 s < t < 480 s), the pressure variations show indications of period-2 dynamics, i.e., the minima of the pressure oscillations alternate between a low and a somewhat higher value. It is well known that period-2 dynamics is a typical precursor of deterministic chaos [8]. 

As shown by Bliimel and co-workers [227], the Icicked Csl molecule is particularly appropriate candidate for this study since it has a large dipole moment, which increases the molecule-field coupling strength, and the rotation-vibration coupling is small at low excitation energies so that one may consider solely rotational excitation. We consider then the dynamics of Csl in the indicated pulsed field, in a parameter range known to display classical chaos [231].. ... 

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 ... 

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