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Complex balancing reconstructed to the asymptotic stability of mass-action chemical reaction networks with conservation laws. (English) Zbl 1407.34067

Summary: Motivated by the fact that the pseudo-Helmholtz function is a valid Lyapunov function for characterizing asymptotic stability of complex balanced mass-action systems (MASs), this paper develops the generalized pseudo-Helmholtz function for stability analysis for more general MASs assisted with conservation laws. The key technique is to transform the original network into two different MASs, defined by reconstruction and reverse reconstruction, with an important aspect that the original network is dynamically equivalent to the reverse reconstruction. Stability analysis of the original network is then conducted based on an analysis of how stability properties are retained from the original network to the reverse reconstruction. We prove that the reverse reconstruction possesses at most an equilibrium in each positive stoichiometric compatibility class if the corresponding reconstruction is complex balanced. Under this complex balanced reconstruction strategy, the asymptotic stability of the reverse reconstruction, which also applies to the original network, is thus reached by taking the generalized pseudo-Helmholtz function as the Lyapunov function. To facilitate applications, we further provide a systematic method for computing complex balanced reconstructions assisted with conservation laws. Some representative examples are presented to exhibit the validity of the complex balanced reconstruction strategy.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
80A30 Chemical kinetics in thermodynamics and heat transfer
93D20 Asymptotic stability in control theory
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92B20 Neural networks for/in biological studies, artificial life and related topics
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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