*Forma,* Vol. 18 (No. 2), pp. 67-82, 2003

*Original Paper*

## The Number of Autocatalytic Reactions in Systems of Oscillating Reaction

Hiroki Takada^{1}*, Yuuki Shimizu1 and Masaru Miyao^{2}

^{1}Graduate School of Science, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

^{2}Information Technology Center, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan

*E-mail address: takada@bio.phys.nagoya-u.ac.jp

(Received November 26, 2002; Accepted March 19, 2003)

**Keywords: **
Belousov-Zhabotinsky (BZ) Reaction, Autocatalytic Reaction, Continuous-Flow Stirred Tank Reactor (CSTR), Equilibrium Space, Feedback System

**Abstract. **
Some mathematical models have been proposed to elucidate the mechanism by which an oscillating reaction is generated as a complex system. Brusselator and Oregonator do not only describe oscillating reaction systems but also reproduce multiple periodic oscillations, burst waves and drawing synchronization. All theoretical researches on these models did not give each differential equation with consideration of the strict correspondence to the system of reactions. Then, we did not adopt a theoretical framework of physics but a mathematical one in which it is possible to survey exhaustively although most of the elements might look apart from real chemical systems. We have been interested in the problem of the least number of chemical species and elementary reactions that can generate the reaction. In this study, we mainly analyzed two chemical species that are not supplied continuously in *Continuous-flow Stirred Tank Reactor (CSTR)*. We have demonstrated that the *oscillating reaction systems* must be described by more than one reaction formula that contains formulae for the *autocatalytic reactions* with physical consideration of the *equilibrium space* of its rate equations. We have contemplated the mathematical meaning of the *autocatalytic reaction* and showed that a positive feedback system gives it; therefore, an *oscillating reaction* can be produced by a complex feedback system.

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