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Chaos, Solitons & Fractals

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Lapacian Frameworks -- Mathematics


The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances. In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment of the so-called defi- ciency zero systems of CRN theory. We show that much of the dynamics of these polynomial systems of differential equations can be understood by analyzing the directed graph Laplacian associated with the system. Beside the more concise mathematical treatment, this leads to con- siderably stronger results. In particular, (i) we show that our Laplacian deficiency zero theorem is markedly stronger than the traditional one and (ii) we derive simple equations for the locus of the equilibria in all (Laplacian) deficiency zero cases.


© Copyright the author(s) 2023


This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: Veerman, J. J. P., Whalen-Wagner, T., & Kummel, E. (2023). Chemical reaction networks in a Laplacian framework. Chaos, Solitons & Fractals, 166, 112859.

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