[Paper Review] A note on the top Lyapunov exponent of linear cooperative systems

Michel Benaı̈m, Claude Lobry+2 more|arXiv (Cornell University)|Feb 12, 2023

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper extends a recent asymptotic formula for the top Lyapunov exponent of linear cooperative systems by replacing the assumption of T-periodic environmental switching with a general uniquely ergodic Feller Markov process. It derives explicit asymptotic expressions for the top Lyapunov exponent in both fast (T→∞) and slow (T→0) environmental fluctuation regimes, establishing convergence to a limit involving the invariant measure and the dominant eigenvector of the averaged system.

ABSTRACT

In a recent paper [Asymptotic of the largest Floquet multiplier for cooperative matrices Annales de la Faculté des Sciences de Toulouse, Tome XXXI, no 4 (2022)] P. Carmona gives an asymptotic formulae for the top Lyapunov exponent of a linear T-periodic cooperative differential equation, in the limit T goes to infinity. This short note discusses and extends this result.

Motivation & Objective

• To generalize a recent asymptotic formula for the top Lyapunov exponent of linear cooperative systems from periodic to more general environmental switching processes.
• To analyze the behavior of the top Lyapunov exponent in the limits of fast (T→∞) and slow (T→0) environmental fluctuations.
• To establish the convergence of the invariant measure of the coupled (environment, population state) process under time-rescaling, and characterize its limit.
• To extend the applicability of Lyapunov exponent asymptotics to non-periodic, uniquely ergodic Markov processes, including stochastic and deterministic quasi-periodic dynamics.

Proposed method

• The system is modeled as a linear differential equation dy/dt = A(ωₜ)y, where ωₜ is a continuous-time, uniquely ergodic Feller Markov process on a compact metric space S.
• The top Lyapunov exponent Λ is defined as the long-term growth rate of the population norm, expressed as the integral of ⟨A(s)θ, 1⟩ over the invariant measure π of the (ωₜ, θₜ) process.
• The paper analyzes the rescaled process ωₜ^T = ω_{t/T}, which models environmental fluctuations at time scale T, and studies the limit of the top Lyapunov exponent Λ^T as T→∞ and T→0.
• It uses the Random Perron-Frobenius theorem and the theory of Feller processes to prove weak* convergence of the invariant measures π^T of the rescaled system to a limit measure π^∞ = μ ⊗ δ_{θ*(s)}.
• The convergence of the Markov semigroups Qₜ^T to Qₜ^∞ is established via uniform continuity and Feller continuity, ensuring convergence of the invariant measures.
• The key technical step involves bounding the difference between the flow under ωₜ^T and the flow under a fixed s, using estimates on the derivative of the flow with respect to initial conditions.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the top Lyapunov exponent when the environmental process ωₜ is rescaled to ωₜ^T = ω_{t/T} as T→∞?
• RQ2 How does the top Lyapunov exponent behave in the limit T→0, corresponding to slowly varying environments?
• RQ3 Under what conditions does the invariant measure of the (environment, population state) process converge as T→∞ or T→0?
• RQ4 Can the asymptotic formula for the top Lyapunov exponent be extended beyond periodic environmental switching to general uniquely ergodic Markov processes?

Key findings

• As T→∞, the top Lyapunov exponent Λ^T converges to the integral of the dominant eigenvalue of A(s) over the invariant measure μ of the environment, i.e., Λ^∞ = ∫_S λ_1(A(s)) dμ(s).
• As T→0, the top Lyapunov exponent Λ^T converges to the average of the dominant eigenvalues of A(s) over the environment, i.e., Λ^0 = ∫_S λ_1(A(s)) dμ(s), which matches the limit in the fast regime.
• The invariant measure π^T of the rescaled system (ωₜ^T, θₜ) converges weak* to μ ⊗ δ_{θ*(s)} as T→∞, where θ*(s) is the globally attracting state of the system with fixed A(s).
• The convergence of the Markov semigroups Qₜ^T to Qₜ^∞ is uniform in s∈S, which ensures the convergence of the invariant measures and the Lyapunov exponent.
• The results hold under general conditions: A(s) is Metzler and irreducible for μ-almost every s, and (ωₜ) is a uniquely ergodic Feller Markov process on a compact metric space.