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Universal Lyapunov functions for non-linear reaction networks
journal contributionposted on 2019-08-30, 14:10 authored by Alexander N. Gorban
In 1961, Rényi discovered a rich family of non-classical Lyapunov functions for kinetics of the Markov chains, or, what is the same, for the linear kinetic equations. This family was parameterized by convex functions on the positive semi-axis. After works of Csiszár and Morimoto, these functions became widely known as f-divergences or the Csiszár–Morimoto divergences. These Lyapunov functions are universal in the following sense: they depend only on the state of equilibrium, not on the kinetic parameters themselves. Despite many years of research, no such wide family of universal Lyapunov functions has been found for nonlinear reaction networks. For general non-linear networks with detailed or complex balance, the classical thermodynamics potentials remain the only universal Lyapunov functions. We constructed a rich family of new universal Lyapunov functions for any non-linear reaction network with detailed or complex balance. These functions are parameterized by compact subsets of the projective space. They are universal in the same sense: they depend only on the state of equilibrium and on the network structure, but not on the kinetic parameters themselves. The main elements and operations in the construction of the new Lyapunov functions are partial equilibria of reactions and convex envelopes of families of functions.
The work was supported by the University of Leicester and the Ministry of Science and Higher Education of the Russian Federation (Project No. 14.Y26.31.0022).
CitationCommunications in Nonlinear Science and Numerical Simulation, 2019, 79, 104910
Author affiliation/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics
- AM (Accepted Manuscript)