[Paper Review] Estimates for principal Lyapunov exponents: A survey

Janusz Mierczyński|arXiv (Cornell University)|Jun 4, 2014

Differential Equations and Boundary Problems23 references6 citations

TL;DR

This survey provides estimates for the principal Lyapunov exponent of time-dependent linear differential equations with monotonicity and order-preserving properties, focusing on strongly cooperative ODEs and parabolic PDEs. It establishes that the principal Lyapunov exponent is bounded by the spectral radius of time-averaged or state-averaged systems, with key results derived via ergodic theory and matrix inequalities, showing temporal variation can enhance population persistence.

ABSTRACT

This is a survey of known results on estimating the principal Lyapunov exponent of a time-dependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further research are hinted.

Motivation & Objective

• To survey known estimates for the principal Lyapunov exponent in time-dependent linear systems with strong order-preserving properties.
• To analyze how the principal Lyapunov exponent relates to the spectral radius of time-averaged or derived time-independent systems.
• To unify results across ODEs, PDEs, and discrete-time systems under a common framework of positivity and monotonicity.
• To identify open problems in estimating the principal Lyapunov exponent via averaging techniques in both continuous and discrete time.
• To highlight implications for population dynamics, where a positive principal Lyapunov exponent ensures long-term population persistence.

Proposed method

• Utilizes the spectral mapping theorem and the principle of linearized stability for nonautonomous systems.
• Applies Birkhoff's ergodic theorem to represent the principal Lyapunov exponent as a time average of logarithmic growth rates.
• Employs geometric-arithmetic mean inequalities to derive bounds on the spectral radius of product matrices.
• Analyzes the transition matrix $U(T;0)$ for periodic systems and relates its spectral radius to the principal Lyapunov exponent.
• Uses positivity and irreducibility of the system to ensure existence of a unique positive solution whose growth rate defines the principal Lyapunov exponent.
• Applies functional-analytic tools, including evolution semigroups on $L_p$ or $C_0$ spaces, to study order-preserving semigroups.

Experimental results

Research questions

• RQ1 How can the principal Lyapunov exponent of a time-periodic cooperative system be estimated using time-averaged coefficients?
• RQ2 What is the relationship between the spectral radius of the monodromy matrix $U(T;0)$ and the exponential growth rate of the principal solution?
• RQ3 In what cases does the principal Lyapunov exponent exceed the principal eigenvalue of the time-averaged system, and when is equality achieved?
• RQ4 Can a unified theory be developed to estimate the principal Lyapunov exponent across ODEs, PDEs, and nonlocal dispersal equations via averaging?
• RQ5 How does temporal variation in system parameters affect the persistence of solutions, particularly in population models?

Key findings

• The principal Lyapunov exponent $\Lambda$ satisfies $\Lambda \geq \log \rho(A\overline{B})$, where $A\overline{B}$ is the matrix formed from time-averaged coefficients, with equality possible in non-degenerate cases.
• For periodic systems, the principal Lyapunov exponent is bounded above by the logarithmic growth rate of the solution operator, which is determined by the spectral radius of $U(T;0)$.
• The sequence $\widetilde{w}_i(n)$ of time-averaged weights converges to a positive vector $\overline{w}$ satisfying $A\overline{B}\overline{w} \leq e^{\Lambda}\overline{w}$, implying $\log \rho(A\overline{B}) \leq \Lambda$.
• The estimates are derived via ergodic theory, ensuring $\widetilde{\mu}(n)$ remains bounded away from zero, which supports convergence of time averages.
• In population models, a positive principal Lyapunov exponent ensures permanence, and temporal variation can enhance persistence, contrary to classical intuition.
• The functional-analytic approach via evolution semigroups on $L_p$ or $C_0$ spaces provides a powerful tool for studying order-preserving systems, especially in periodic settings.