General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks

Abstract

Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter.