Lyapunov Exponents for Sparsely Coupled Linear Cocycles

TL;DR

This paper develops methods to compute or bound the top Lyapunov exponent for structured products of matrices with zero patterns, including sparse and block-triangular matrices, applicable to deterministic and stochastic systems.

Contribution

It introduces explicit bounds and formulas for the top Lyapunov exponent using a generalized triangular reduction and extends these results to ergodic and perturbation models.

Findings

Explicit bounds for Lyapunov exponents in structured matrix products

Formulas for top Lyapunov exponent in favorable cases

Applicability to ergodic cocycles and perturbation models

Abstract

This paper studies structured products of real matrices for which the top Lyapunov exponent can be accessed by reducing the dynamics to an amenable generalization of upper triangular matrices. Exploiting prescribed zero patterns (including block-triangularity and sparse decompositions, conveniently encoded by a directed sparsity graph), we obtain explicit, computable bounds and, in favorable cases, formulas for $\gamma_1$ by combining deterministic triangular controls with a suitable refinement of the Furstenberg--Kifer lemma for block-triangular products. The estimates apply both to tempered (possibly deterministic) sequences and to stationary ergodic random cocycles under standard integrability. We also discuss applications to perturbation models for linear systems, including low-rank updates, where the reduction converts the problem to lower-dimensional or scalar cocycles.