Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lyapunov exponents of renewal equations: numerical approximation and convergence analysis

Dimitri Breda, Davide Liessi

TL;DR

The paper develops a rigorous, convergent methodology for computing Lyapunov exponents of renewal equations by formulating the problem in a Hilbert space $L^{2}$, discretizing the associated evolution operators with a pseudospectral approach, and applying a QR-based scheme to extract exponents. It proves norm convergence of the discretized operators to the continuous evolution, and provides a backward-error framework showing convergence of the computed Lyapunov spectra to the exact ones under suitable assumptions. The authors implement the method in MATLAB and validate it on a renewal equation with a quadratic nonlinearity, reproducing known dynamical transitions such as Hopf bifurcations, period-doubling, and chaos, in agreement with prior results. This work provides a rigorous, practical tool for stability and chaotic dynamics analysis in renewal/Volterra-type delay models relevant to population dynamics and related fields.

Abstract

We propose a numerical method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type), consisting first in applying a discrete QR technique to the associated evolution family suitably posed on a Hilbert state space and second in reducing to finite dimension each evolution operator in the obtained time sequence. The reduction to finite dimension relies on 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.

Lyapunov exponents of renewal equations: numerical approximation and convergence analysis

TL;DR

The paper develops a rigorous, convergent methodology for computing Lyapunov exponents of renewal equations by formulating the problem in a Hilbert space , discretizing the associated evolution operators with a pseudospectral approach, and applying a QR-based scheme to extract exponents. It proves norm convergence of the discretized operators to the continuous evolution, and provides a backward-error framework showing convergence of the computed Lyapunov spectra to the exact ones under suitable assumptions. The authors implement the method in MATLAB and validate it on a renewal equation with a quadratic nonlinearity, reproducing known dynamical transitions such as Hopf bifurcations, period-doubling, and chaos, in agreement with prior results. This work provides a rigorous, practical tool for stability and chaotic dynamics analysis in renewal/Volterra-type delay models relevant to population dynamics and related fields.

Abstract

We propose a numerical method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type), consisting first in applying a discrete QR technique to the associated evolution family suitably posed on a Hilbert state space and second in reducing to finite dimension each evolution operator in the obtained time sequence. The reduction to finite dimension relies on 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.
Paper Structure (12 sections, 13 theorems, 82 equations, 3 figures)

This paper contains 12 sections, 13 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Renewal equations on L2
  3. Evolution operators
  4. Definition and compactness
  5. Discretization
  6. Convergence
  7. Lyapunov exponents
  8. Definition
  9. Approximation
  10. Convergence
  11. Implementation and numerical results
  12. Concluding remarks

Key Result

Theorem 2.1

If $K\in L^{\infty}(J^2,\mathbb{R}^{d\times d})$ and $g\in L^p(J,\mathbb{R}^{d})$ then VIEVIEVIE has a unique solution $x\in L^p(J,\mathbb{R}^{d})$ on $J$, given by the variation of constant formula where $R$ is the resolvent of $K$ on $J$.

Figures (3)

  • Figure 1: Absolute errors on the dominant LEs of the RE with quadratic nonlinearity \ref{['quadRE']}quadREquadRE 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 eigTMNc. The final time is $t_{\text{f}}=1000$.
  • Figure 2: Absolute errors on the dominant LEs of the RE with quadratic nonlinearity \ref{['quadRE']}quadREquadRE 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 eigTMNc. The exponents are computed for $M=N=16$.
  • Figure 3: Diagram of the first two dominant (in descending order) LEs of the RE with quadratic nonlinearity \ref{['quadRE']}quadREquadRE when varying $\gamma$, computed with $M=N=15$ and $t_{\text{f}}=1000$.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.2: BredaLiessi2018
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 12 more