arXiv subjects

Davide Liessi

Publications and source records attributed to Davide Liessi.

8 recordsLinked to original sources

arXiv2025

Approximating evolution operators of linear delay equations: a general framework for the convergence analysis

We consider the problem of discretizing evolution operators of linear delay equations with the aim of approximating their spectra, which is useful in investigating the stability properties of (nonlinear) equations via the principle of linearized stability. We develop a general convergence analysis based on a reformulation of the operators by means of a fixed-point equation, providing a list of hypotheses related to the regularization properties of the equation and the convergence of the chosen approximation techniques on suitable subspaces. This framework unifies the proofs for some methods based on pseudospectral discretization, which we present here in this new form. To exemplify the generality of the framework, we also apply it to a method of weighted residuals found in the literature, which was previously lacking a formal convergence analysis.

math.NA↗

arXiv2025

New functionalities in MatCont: delay equations and Lyapunov exponents

MatCont is a powerful toolbox for numerical bifurcation analysis focussing on smooth ODEs. A user can study equilibria, periodic and connecting orbits, and their stability and bifurcations. Here, we report on additional features in version 7p6. The first is a delay equation importer enabling MatCont users to study a much larger class of models, namely delay equations with finite delay (including delay differential and renewal equations). This importer translates the delay equation into a system of ODEs using a pseudospectral approximation with an order specified by the user. We also implemented Lyapunov exponent computations, event functions for Poincar\'e maps, and enhanced homoclinic continuation. We demonstrate these features with test cases, such as the Mackey-Glass equation and a renewal equation, and provide additional examples in online tutorials.

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arXiv2024

Lyapunov exponents of renewal equations: numerical approximation and convergence analysis

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.

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arXiv2023

A practical approach to computing Lyapunov exponents of renewal and delay equations

We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.

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arXiv2023

Spectra of evolution operators of a class of neutral renewal equations: theoretical and numerical aspects

In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear periodic equation with one discrete delay and fully characterize the spectrum of its monodromy operator. We perform numerical experiments discretizing the evolution operators via pseudospectral collocation, confirming the theoretical results and giving perspectives on the generalization to systems and to multiple delays. Although we do not attempt to perform a rigorous numerical analysis of the method, we give some considerations on a possible approach to the problem.

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arXiv2022

A practical guide to piecewise pseudospectral collocation for Floquet multipliers of delay equations in MATLAB

In recent years we provided numerical methods based on pseudospectral collocation for computing the Floquet multipliers of different types of delay equations, with the goal of studying the stability of their periodic solutions. The latest work of the series concerns the extension of these methods to a piecewise approach, in order to take the properties of numerically computed solutions into account. In this chapter we describe the MATLAB implementation of this method and provide practical usage examples.

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arXiv2022

Piecewise discretization of monodromy operators of delay equations on adapted meshes

Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.

math.NA↗

arXiv2022

A pseudospectral method for investigating the stability of linear population models with two physiological structures

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. We propose an equivalent reformulation of the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

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