Oscillatory reactions chaotic oscillations

IIIC) 1978 Wegmann, K., Rossler, O. E. Different Kinds of Chaotic Oscillations in the Belousov-Zhabotinskii Reaction, Z. Naturforschung A, vol. 33A, no. 10, 1170-1183 (III J) 1980 Willamowski, Rossler, O. E. Irregular Oscillations in a Realistic Abstract Quadratic Mass Action System, Z, Naturforsch., vol. 35a, 317-318 (IIIG) 1965 Yamazaki, L., Yokoya, K., Nakajima, R. Oscillatory Oxidations of Reduced Pyridine Nucleotide by Peroxidase, Biochim. Biophys. Res. Commun., vol. 21, 582-586 (IIIG) 1967 Yamazaki, I., Yokota, K. Analysis of the Conditions Causing the Oscillatory Oxidation of Reduced Nicotinamide-Adenine Dinucleotide by Biochem. Biophys. Acta, vol. 132, 310-320... 

In spite of well-developed classical theory, the interest to investigation of synchronization phenomena essentially increased within last two decades and this discipline still remains a field of active research, due to several reasons. First, a discovery and analysis of chaotic dynamics in low-dimensional deterministic systems posed a problem of extension of the theory to cover the case of chaotic oscillators as well. Second, a rapid development of computer technologies made a numerical analysis of complex systems, which still cannot be treated analytically, possible. Finally, a further development of synchronization theory is stimulated by new fields of application in physics (e.g., systems of coupled lasers and Josephson junctions), chemistry (oscillatory reactions), and in biology, where synchronization phenomena play an important role on all levels of organization, from cells to physiological subsystems and even organisms. 

In this section the whole field of exotic dynamics is considered this term includes not merely oscillating reactions but also oligo-oscillatory reactions, multiple steady states, spatial phenomena such as travelling reaction waves, and chaotic systems. All of these have common roots in autocatalytic processes. This area has continued to expand, and there is a case for treatment in future volumes by a specialist reviewer. An entry into the literature can be gained from a recent series of articles in a chemical education joumal, and in a festschrift issue in honor of Professor R. M. Noyes. Other useful sources are a volume of conference proceedings, and a volume of lecture preprints of a 1989 conference. The present summary is concerned with the chemical rather than the mathematical aspects of the topic. 

In order to understand the complexity in oscillatory reactions, it would be worthwhile to examine their relationship with different types of chemical reactions [10], which have been summarized in Fig. 9.8 in increasing order of complexity viz., irreversible reactions -> reversible reactions parallel reaction consecutive reaction -> autocatalytic reactions damped oscillations aperiodic oscillations spatio-temporal oscillations chaotic oscillations. Further, Fig. 9.8 shows the concentration... 

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

Are these phenomena unique, or are they typical of biological systems From a mathematical perspective, enzyme systems fall into a class of nonlinear organization, and a chain of enzyme reactions with negative feedback easily can demonstrate oscillatory behavior [520]. Glass has noted that in general, any nonlinear system with multiple negative feedback may demonstrate oscillations that lead to chaotic behavior [595]. 

Some of the oscillatory solutions, in particular those found in the abstract models of Rossler, are attractors, moreover they are chaotic. These mathematical solutions are interesting in that many oscillations observed experimentally are probably of this nature, and if studied with models encompassing the true behavior of the reaction they can be obtained theoretically. These chaotic attractors are illustrated by the examples given in Section III.I. 

Using an opposite approach, that of starting from a mathematical solution and designing an experiment, Olsen and Degn (1972) showed that abstract models may lead to an understanding of the oscillatory chemical reactions exhibiting not only just limit cycle oscillations but also chaotic attractor-type oscillations. 

It should be noted that chaos control can only be obtained if deterministic chaos is involved. In case of (i) chaotic laser (ii) diode (iii) hydrodynamic and magneto-elastic systems and (iv) more recently myocardial tissue, feedback algorithm has been successfully applied to stabilize periodic oscillations. Quite recently, in order to stabilize periodic behaviour in the chaotic regime of oscillatory B-Z reaction, Showalter [14] and co-workers (1998) applied proportional feedback mechanism. Feedback was applied to the system by perturbing the flow rate of cesium-bromate solutions in the reactor keeping the flow rate of malonic acid fixed in these experiments. This experimental arrangement helped the stabilization of periodic behaviour within the chaotic regime. 

Various types of oscillating behaviors such as emergence of chemical waves, chaotic patterns, and a rich variety of spatiotemporal structures are investigated in oscillatory chemical reactions in association with nonlinear chemical dynamics [1-3]. In non-equilibrium condition, the characteristic dynamics of such chemically reacting systems are capable to self-organize into diverse kinds of assembly patterns. With the help of nonlinear chemical dynamics, the complexity and orderliness of those chemical processes can be explained properly. Various biological processes which exhibited very time-based flucmations especially when they are away from equilibrium have also been described by mechanistic considerations and theoretical techniques of nonlinear chemical dynamics [4-7]. 

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