Entropy-inspired Lyapunov Functions and Linear First Integrals for Positive Polynomial Systems

Abstract

Two partially overlapping classes of positive polynomial systems, chemical reaction networks with mass action law (MAL-CRNs) and quasi-polynomial systems (QP systems) are considered. Both of them have an entropy-like Lyapunov function associated to them which are similar but not the same. Inspired by the work of Prof. Gorban [12] on the entropy-functionals for Markov chains, and using results on MAL-CRN and QP-systems theory we characterize MAL-CRNs and QP systems that enable both types of entropy-like Lyapunov functions. The starting point of the analysis is the class of linear weakly reversible MAL-CRNs that are mathematically equivalent to Markov chains with an equilibrium point where various entropy level set equivalent Lyapunov functions are available. We show that non-degenerate linear kinetic systems with a linear first integral (that corresponds to conservation) can be transformed to linear weakly reversible MAL-CRNs using linear diagonal transformation, and the coefficient matrix of this system is diagonally stable. This implies the existence of the weighted version of the various entropy level set equivalent Lyapunov functions for non-degenerate linear kinetic systems with a linear first integral. Using translated X-factorable phase space transformations and nonlinear variable transformations a dynamically similar linear ODE model is associated to the QP system models with a positive equilibrium point. The non-degenerate kinetic property together with the existence of positive equilibrium point form a sufficient condition of the existence of the weighted version of the various entropy level set equivalent Lyapunov functions in this case. Further extension has been obtained by using the time re-parametrization transformation defined for QP models.