Chaos, chemical deterministic

Chaos in deterministic systems strange attractors, turbulence, and applications in chemical engineering. Chem. Eng. Sci., 43, 139-183. 

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 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],... 

In the nonthermodynamic branch at the region of unstable stationary states, the system properties depend on the particular form of differential equations to describe its dynamic behavior when the parameters are behind the bifurcation point. For example, the system behavior may be like a chemical machine, which is strongly deterministic by starting conditions, or it may correspond to the so called chaos when any infin itesimal fluctuations cause heavy and irregular changes in the system states. 

At the microscopic level, chemical reactions are dynamical phenomena in which nonlinear vibrational motions are strongly coupled with each other. Therefore, deterministic chaos in dynamical systems plays a crucial role in understanding chemical reactions. In particular, the dynamical origin of statistical behavior and the possibility of controlling reactions require analyses of chaotic behavior in multidimensional phase space. 

Taken together, these results demonstrate that deterministic chaos can occur in a nonequilibrium chemical system. The most remarkable thing is that the re-... 

Thus, 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. 

The main contribution of non-linear chemistry is the pitchfork bifurcation diagram. A stable state becomes uixstable and bifurcates into two new stable branches. We are unable to foresee which one of these states will be chosen by the nature of the physico-chemical reaction. The multiplicity of choices gives its full importance to Ae evolution of the systems. This paper has aimed to show the extent to which the concepts of non-equilibrium and of deterministic chaos sublimate the fundamental physical laws by leading us to the creation of new structures and to auto-organization. Chemistry is no exception to this rule. The final conclusion is given by Jean-Marie Lehn [21], Nobel Prize Winner in... 

In addition to the multiple steady states and sustained oscillations, chemical reactors are a rich source of other types of behavior including deterministic chaos [18, 17, 23, 14], The chemical engineer should be aware of the complex behavior that is possible with simple nonlinear CSTR models, especially if confronted with apparently complex operating data. 

Stewart, 1989). In many respects, the idea that systems with a deterministic dynamics can behave in ways that we normally associate with systems subject to random forces—that identical experiments on macroscopic systems can lead to very different results because of tiny, unavoidable differences in initial conditions— is truly revolutionary. Chaos has become a more central part of mathematics and physics than of chemistry, but it is now clear that chemical systems also exhibit chaotic behavior (Scott, 1991). In this section, we define chaos and some related concepts. We also give some examples of chaotic chemical systems. In the next section, we discuss the intriguing notion that understanding chaos may enable us to control chaotic behavior and tailor it to our needs. 

Although there are many definitions of chaos (Gleick, 1987), for our purposes a chaotic system may be defined as one having three properties deterministic dynamics, aperiodicity, and sensitivity to initial conditions. Our first requirement implies that there exists a set of laws, in the case of homogeneous chemical reactions, rate laws, that is, first-order ordinary differential equations, that govern the time evolution of the system. It is not necessary that we be able to write down these laws, but they must be specifiable, at least in principle, and they must be complete, that is, the system cannot be subject to hidden and/or random influences. The requirement of aperiodicity means that the behavior of a chaotic system in time never repeats. A truly chaotic system neither reaches a stationary state nor behaves periodically in its phase space, it traverses an infinite path, never passing more than once through the same point. 

However, at that time there still existed considerable healthy skepticism regarding the existence of nonperiodic behavior in well-controlled nonequilibrium chemical reactions. After all, nonperiodic behavior can arise from fluctuations in stirring rate or flow rate, evolution of gas bubbles from the reaction, spatial inhomogeneities due to incomplete mixing, vibrations in the stirring motor, fluctuations in the amount of bromide and dissolved oxygen in the feed, and so on. Any experimental data, no matter how well a system is controlled, will contain some noise arising from fluctuations in the control parameters therefore, it is reasonable to ask "Will noise, always present in experiments, inevitably mask deterministic nonperiodic behavior (chaos) "... 

The answer is no. We will summarize the large body of evidence, gathered recently by groups in Bordeaux [13-18], Virginia [19-20], and Texas [21-25], that indicates that the nonperiodic behavior observed in chemical reactions is, at least in some cases, chaos, not noise. It is even sometimes possible to separate the experimental noise from the deterministic dynamics. In addition, as we shall describe, low-dimensional deterministic models deduced from the data make it possible to predict the behavior that should occur as a control parameter (e.g., flow rate) is varied, and some of these predictions have been confirmed by experiment. 

Each chaotic state C, consists primarily of what at first glance appears to be a stochastic mixture of the adjacent periodic states and thus, for example, consists of a mixture of Po and P, as Fig. 6 of ROUX et al. [23] illustrates. However, maps constructed from the time series clearly yield smooth curves, not a scatter of points. These maps indicate that behavior is deterministic, not stochastic. Moreover, it is difficult to imagine stochastic processes that would lead to period doubling, the universal sequence, and tangent bifurcations, yet all of these phenomena associated with chaos are found in one-dimensional maps and in experiments on nonequilibrium chemical reactions, as will now be described. 

Abstract chemical models exhibiting nonlinear phenomena were proposed more than a decade ago. The Brusselator of PRIGOGINE and LEFEVER [54] has oscillatory (limit cycle) solutions, and the SCHLOGL [55] model exhibits bistability, but these models have only two variables and hence cannot have chaotic solutions. At least 3 variables are required for chaos in a continuous system, simply because phase space trajectories cannot cross for a deterministic system. As mentioned in the Introduction, the possibility of chemical chaos was suggested by RUELLE [1] in 1973. In 1976 ROSSLER [56], inspired by LORENZ s [57] study of chaos in a 3 variable model of convection, constructed an abstract 3 variable chemical reaction model that exhibited chaos. This model used as an autocatalytic step a Michaelis-Menten type kinetics, which is a nonlinear approximation discovered in enzymatic studies. Recently more realistic biochemical models [58,59] have also been found to exhibit low dimensional chaos. 

As a direct consequence of the particular role of Dynamics, as such,in the study of non-equilibrium behaviour of chemical systems, two classes of models are to be considered, depending on which aspect one is insisting on. Formal models, of mathematical or chemical-like nature, are designed to exhibit specific dynamical behaviours, without too much concern about chemical significance. Their aim is to provide examples of evolution equations of chemical reacting systems, as described by mass action kinetics, that are able to produce those exotic behaviours, such as bistability or multistability, between various types of attractors, like steady states, oscillations or deterministic chaos. A typical historical model of that kind is the "Brusselator ... 

Chemical dynamics in a well-stirred reactor provides one of the most clearcut examples of deterministic chaos, since it can generate this type of behavior from the intrinsic nonlinearities of the reaction mechanism rather than from the spatial degrees of freedom as it happens in hydrodynamics. This form of chaos is amenable to a small number of variables and constitutes, therefore. 

If chemical wave propagation is studied in the vicinity of chaos one has the possibility that internal noise can lead to perturbations of the dynamics that cause the system to behave as if it were in nearby period-doubled or even chaotic regions. Such local fluctuations can then give rise to new wave patterns that are not observed in deterministic systems. These effects are the spatial analogs of the noisy bifurcation shifts seen in systems described by ODEs or maps [23, 24]. 

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