Chaos, deterministic

Deterministic chaos is a special type of chaos, which can be predicted on the basis of a set of differential equations. In the next section, deterministic chaos will be discussed in detail. 

The phenomenon of deterministic chaos was established by the work of Lorenz who showed for the first time that chaos is not unnecessarily an unpredictable phenomenon. A certain class can be governed by mathematical equations suggesting it s predictability. This type of chaos is called deterministic chaos. 

We give below some typical examples of deterministic equations which yield chaotic oscillations and specific class of trajectory called Strange attractor on computer solution  

For (T = 10, b = 8/3, Lorenz found that the set of equations yield chaotic oscillations time series whenever r exceeds a critical value r 24.74. The Lorenz attractor can be reconstructed fi om a time series with a delay time (t = 0.1). 

The phase-plane plot in three dimensions is further shown in Fig. (12.8). The phase-plane plot in three dimensions has fractal geometry and the attractor is called the strange attractor. 

Recent studies have indicated that fluidized beds may be deterministic chaotic systems (Daw etal.,1990 Daw and Harlow, 1991 Schouten and van den Bleek, 1991 van den Bleek and Schouten, 1993). Such systems are characterized by a limited ability to predict their evolution with time. If fluidized beds are deterministic chaotic systems, the scaling laws should reflect the restricted predictability associated with such systems. 

Further work is needed to determine in which regimes, if any, fluid bed behave as chaotic systems. Additional testing is needed to determine the sensitivity of important bed hydrodynamic characteristics to the Kolmogorov entropy, to quantitatively relate changes of entropy to 

Back to chemistry. We have already shown that the Brusselator leads to the existence of a limit cycle. In more elaborate models, this cycle can be subdivided into various periodicities to eventually give rise to a strange attractor. 

Several groups have shown the existence of a strange attractor in the BZ reaction. The representation below was taken from the work of Gyorgi, Rempe and Field [14]. 

it is well established that deterministic chaos plays a role in chemistry. It has been analyzed in different chemical processes. Asked about the importance of chemical oscillations and chaos in the chemistry of mass industrial production, Wasserman, former director of research at Dupont de Nemours and past-president of the American Chemical Society in 1999, said [15] Fes. The new tools of nonlinear dynamics have allowed us a fresh viewpoint on reactions of interest . Ehipont has identified chaotic phenomena in reactions as important as the conversion of the p-xylol in terephtalic acid or the oxidation of the benzaldehyde in benzoic acid. 

For the interested reader, various theoretical [16] and experimental [17] models are described in the literature related to these non-linear complex behaviors. 

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Gyorgyi L and Field R J 1992 A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction Nature 355 808-10... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

According to Galilei, the observation of natural phenomena using suitable measuring instruments provides certain numerical values which must be related to one another the solution of the equations derived from the numbers allows us to forecast future developments. This led to the misunderstanding that knowledge could only be obtained in such a manner. The result was deterministic belief, which was disproved for microscopic objects by Heisenberg s uncertainty principle. On the macroscopic scale, however, it appeared that the deterministic approach was still valid. Determinism was only finally buried when deterministic chaos was discovered. 

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

Daw, C. S., and Harlow, J. S., Characteristics of Voidage and Pressure Signals from Fluidized Beds using Deterministic Chaos Theory, Proc. 11th Int. Conf. FluidizedBedComb., 2 777 (1991)... 

The last implies that high sensitiveness of solution to the choice of initial conditions, or equivalently deterministic chaos. 

S. Nicolay, E.-B. Brodie of Brodie, M. Touchon, Y. d Aubenton-Carafa, C. Thermes, and A. Arneodo, Erom scale invariance to deterministic chaos in DNA sequences towards a deterministic description of gene organization in the human genome. Physica A 342,270-280 (2004). 

Elbert, Thomas, William J. Ray, and Zbignew J. Kowalik. 1993. "Chaos and Physiology Deterministic Chaos in Excitable CeU Assemblies." Physiological Reviews 74 1-45. 

It was assumed that a description of evolution of deterministic systems required a solution of the equations of motion, starting from some initial conditions. Although Poincare [1] knew that it was not always true, this opinion was common. Since the work of Lorenz [2] in 1963, unpredictability of deterministic systems described by differential nonlinear equations has been discovered in many cases. It has been established that given infinitesimally different initial conditions, the outcomes can be wildly different, even with the simplest equations of motion. This feature means the occurrence of deterministic chaos. The literature devoted to this multidisciplinary and rapidly developing discipline of science is huge. There are many excellent textbooks, monographs, and collections of main papers, and we mention only a few [3-8]. 

Nonlinear optics is a very convenient area to investigate the phenomenon of deterministic chaos both from theoretical and experimental points of view. 

H. G. Schuster, Deterministic Chaos. An Introduction, VCH Verlagsgesellschaft, Weinheim, 1988. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

The Lorenz and Rossler models are deterministic models and their strange attractors are therefore called deterministic chaos to emphasize the fact that this is not a random or stochastic behavior. 

Mathematical models that describe and predict the inanimate world quite well are actually of little value in the system of deterministic chaos that governs biology. The answers one can expect from mathematical approaches to evolution (in contrast to my earlier perception) cannot be narrowed to less than the surface of the chaotic attractor of the system which is a little like watching evolution on earth from a satellite.1 The limits of the attractor surface are given by the initial conditions which are not knowable in sufficient detail.2 Empiricism can help, after all our laws of science by and large are the results of repeated observations. 

The description of small scale turbulent fields in confined spaces by fundamental approaches, based on statistical methods or on the concept of deterministic chaos, is a very promising and interesting research task nevertheless, at the authors knowledge, no fundamental approach is at the moment available for the modeling of large-scale confined systems, so that it is necessary to introduce semi-empirical models to express the tensor of turbulent stresses as a function of measurable quantities, such as geometry and velocity. Therefore, even in this case, a few parameters must be adjusted on the basis of independent measures of the fluid dynamic behavior. In any case, it must be underlined that these models are very complex and, therefore, well suited for simulation of complex systems but neither for identification of chemical parameters nor for online control and diagnosis [5, 6],... 

Figure 14.9 Phase portraits of the resonance circuit with a TGS-crystal at different driving voltages below the phase transition and the corresponding Fourier spectra of the response functions with the periods T, 2T, 4T and deterministic chaos...

Before we can start to develop a model we also have to decide how to interpret the behavior observed in Fig. 2.1. The variations in insulin and glucose concentrations could be generated by a damped oscillatory system that was continuously excited by external perturbations (e.g. through interaction with the pulsatile release of other hormones). However, the variations could also represent a disturbed self-sustained oscillation, or they could be an example of deterministic chaos. Here, it is important to realize that, with a sampling period of 10 min over the considered periods of 20-24 h, the number of data points are insufficient for any statistical analysis to distinguish between the possible modes. We need to make a choice and, in the present case, our choice is to consider the insulin-glucose regulation to operate... 

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]. 

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