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Chemical applications of topology and group theory

14. Topological aspects of chaotic chemical reactions

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Abstract

Earlier approaches to the analysis of chemical dynamic systems using kinetic logic are refined to deal more effectively with systems having the two or more feedback circuits required for chaos. The essential kinetic features of such a system can be represented by a directed graph (called aninfluence diagram) in which the vertices represent the internal species and the directed edges represent kinetic relationships between the internal species. Influence diagrams characteristic of chaotic chemical systems have the following additional features: (1) They are connected; (2) Each vertex has at least one edge directed towards it and one edge directed away from it; (3) There is at least one vertex, called a turbulent vertex, with at least two edges directed towards it. From such an influence diagram a state transition diagram representing the qualitative dynamics of the system can be obtained using the following 4-step procedure: (1) A logical relationship is assigned at each turbulent vertex; (2) A local truth table is generated for each circuit in the influence diagram; (3) The local truth tables are combined to give a global truth table using the logical relationships at the turbulent vertices; (4) The global truth table is used to determine the corresponding state transition diagram using previously described methods. This refined procedure leads to a more restricted set of influence diagrams having the interlocking cycle flow topology required for chaos than the procedure described earlier. Systems with 3 internal species are examined in detail using the refined procedure. All systems with 3 dynamic variables shown in the simulation studies of Rössler to give chaotic dynamics correspond to influence diagrams which give inter-locking cycle (chaotic) flow topologies by the refined procedure. In addition, two models for the Belousov-Zhabotinskii reaction are examined using the refined procedure. The results are potentially informative concerning possible mechanisms for the limitation of the accumulation of autocatalytically produced HBrO2 (one of the internal species) during the course of this reaction.

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For part 13 of this series see R. B. King, Theoret. Chim. Acta (Berl.) 63, 103–132 (1983)

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King, R.B. Chemical applications of topology and group theory. Theoret. Chim. Acta 63, 323–338 (1983). https://doi.org/10.1007/BF01151610

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Key words

  • Dynamic systems
  • Chaotic chemical reactions
  • Kinetic logic
  • Turbulence
  • Belousov-Zhabotinskii reaction