Lyapunov exponents of renewal equations: Numerical approximation and convergence analysis

Dimitri Breda; Davide Liessi

doi:10.3934/dcdsb.2024152

Article Contents

2025, Volume 30, Issue 6: 1930-1951. Doi: 10.3934/dcdsb.2024152

This issue Previous Article Construction of a contraction metric for non-smooth time-periodic systems Next Article Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups

Lyapunov exponents of renewal equations: Numerical approximation and convergence analysis

Dimitri Breda and
Davide Liessi^,

CDLab – Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy

^*Corresponding author: Davide Liessi
^*Corresponding author: Davide Liessi

Received: March 2024

Revised: October 2024

Early access: October 23, 2024

Published online: October 23, 2024

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Abstract

We propose a numerical method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type), consisting first of applying a discrete QR technique to the associated evolution family suitably posed on a Hilbert state space, and second in reducing to a finite dimension each evolution operator in the obtained time sequence. The reduction to finite dimension relies on a Fourier projection in the state space and on pseudospectral collocation in the forward time step. A rigorous proof of convergence of both the discretized operators and the approximated exponents is provided. A MATLAB implementation is also included for completeness.

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Mathematics Subject Classification: Primary: 37M25, 45D05, 65R20; Secondary: 47D99.

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Figure 1. Absolute errors on the dominant LEs of the RE with quadratic nonlinearity (46) for values of $ \gamma $ corresponding to the stable trivial equilibrium ($ \gamma = 0.5 $), the stable nontrivial equilibrium ($ \gamma = 3 $), and the stable periodic orbit ($ \gamma = 4 $). For the last one, both the trivial and the dominant nontrivial exponents are shown. The errors are measured with respect to the exponents computed via $\texttt{eigTMNc}$. The final time is $ t_{\text{f}} = 1000 $

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Figure 2. Absolute errors on the dominant LEs of the RE with quadratic nonlinearity (46) for values of $ \gamma $ corresponding to the stable trivial equilibrium ($ \gamma = 0.5 $), the stable nontrivial equilibrium ($ \gamma = 3 $), and the stable periodic orbit ($ \gamma = 4 $). For the last one, both the trivial and the dominant nontrivial exponents are shown. The errors are measured with respect to the exponents computed via $\texttt{eigTMNc}$. The exponents are computed for $ M = N = 16 $

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Figure 3. Diagram of the first two dominant (in descending order) LEs of the RE with quadratic nonlinearity (46) when varying $ \gamma $, computed with $ M = N = 15 $ and $ t_{\text{f}} = 1000 $

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