Prigogine and Lefever

Much use has been, and continues to be, made of simplified model schemes representative of general classes of chemical or themial feedback. The oregonator and Lengyel-Epstein models for the BZ and CDIMA systems have been given earlier. Pre-eminent among the more abstracted caricature models is the bnisselator introduced by Prigogine and Lefever [47] which has the following fomi ... 

Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

The next two important steps in this narrative are considered to be the following (i) the description of the bruxellator by Prigogine and Lefever, who, following on from Turing s work, analyzed theoretically the ingredients that should be present in a model of chemical reactions in order to produce spatial self-organization (Prigogine and Lefever, 1968) (ii) the description of the Belousov-Zhabotinsky (B-Z) reaction. 

Prigogine and Lefever introduced a simple two-variable scheme in 1968 [354, 243] that displays sustained oscillatory behavior. It was subsequently dubbed the Brusselator by Tyson [441] and consists of four steps ... 

To illustrate further the analysis of nonlinear differential equations, consider the mechanism due to Prigogine and Lefever (1968)... 

This oscillating system was investigated in 1968 by Prigogine and Lefever. The partial modification in the form of Brusselator was done by Tyson group at Free University of Bmssels in 1973 [30]. The mechanism of this oscillating system along with the rate of equations is properly shown in Table 1.2. 

By analysis and numerical simulation, Prigogine and Lefever demonstrated that their model shows homogeneous oscillations and propagating waves like those seen in the BZ system. The Brusselator was extremely important because it showed that a chemically reasonable mechanism could exhibit self-organization. 

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. 

Prigogine I and Lefever R 1968 Symmetry breaking instabilities in dissipative systems J. Chem. Phys. 48 1695-700... 

In 1969 a paper by I. Prigogine, R. Lefever, A. Goldbveter, and M. Hershkowitz-Kaufman was published Symmetry-breaking instabilities in... 

TNC. 24.1. Prigogine and R. Lefever, Termodinamica e Biologia (Thermodynamics and biology), in Enciclopedia della Scienza e della Tecnica, Mondadori, 1971. 

TNC.34.1. Prigogine and R. Lefever, Theory of dissipative structures, Proceedings, Symposium on Synergetics, Schloss Elmau 1972, H. Haken, ed., Teubner, Stuttgart, 1973, pp.l24—135. 

TNC.39. I. Prigogine and R. Lefever, Stability and thermodynamic properties of dissipative structures in biological systems, in Proceedings, 1st Aharon Katzir-Katchalsky Conference, I. R. Miller, ed., Wiley, New York, 1975, pp. 26-57. 

TNC. 46.1. Prigogine and R. Lefever, Stability and self-organization in open systems, Adv. Chem. Phys. 29, 1-28 (1975). 

TNC.52. 1. Prigogine and R. Lefever, Coupling between difusion and chemical reactions, in 16-e Conseil Solvay de Chimie, 1976, Wiley, New York, 1978, pp. 1—53. 

TNC.65. 1. Prigogine and R. Lefever, On the spatio-temporal evolution of cellular tissues, in Biological Structures and Coupled Flows, A. Oplatka and M. Balaban, eds.. Academic Press, New York, 1983, pp. 3-26. 

I. PRIGOGINE and R. LEFEVER Service de Chimie-Physique II Brussels, Belgium... 

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