49 research outputs found
Numerical approximation of the non-essential spectrum of abstract delay differential equations
Abstract Delay Differential Equations (ADDEs) extend Delay Differential Equations (DDEs) from finite to infinite dimension. They arise in many application fields. From a dynamical system point of view, the stability analysis of an equilibrium is the first relevant question, which can be reduced to the stability of the zero solution of the corresponding linearized system. In the understanding of the linear case, the essential and the non-essential spectra of the infinitesimal generator are crucial. We propose to extend the infinitesimal generator approach developed for linear DDEs to approximate the non-essential spectrum of linear ADDEs. We complete the paper with the numerical results for a homogeneous neural field model with transmission delay of a single population of neurons
Equations with infinite delay: pseudospectral discretization for numerical stability and bifurcation in an abstract framework
We consider nonlinear delay differential and renewal equations with infinite
delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by
introducing a unifying abstract framework and derive a finite-dimensional
approximating system via pseudospectral discretization. For renewal equations,
via integration we consider a reformulation in a space of absolutely continuous
functions that ensures that point evaluation is well defined. We prove the
one-to-one correspondence of equilibria between the original equation and its
approximation, and that linearization and discretization commute. Our most
important result is the proof of convergence of the characteristic roots of the
pseudospectral approximation of the linear(ized) equations, which ensures that
the finite-dimensional system correctly reproduces the stability properties of
the original linear equation if the dimension of the approximation is large
enough. This result is illustrated with several numerical tests, which also
demonstrate the effectiveness of the approach for the bifurcation analysis of
equilibria of nonlinear equations.Comment: 19 pages, 4 figure
Numerical computation of characteristic multipliers for linear time periodic coefficients delay differential equations
In this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to the system of DDEs, i.e. the solution operator over the period of the coefficients, we discretize it by using pseudospectral differencing techniques based on collocation and approximate the dominant multipliers by the eigenvalues of the resulting matrix. The use of pseudospectral methods has already been proposed in the context of simpler DDEs. Here we fully generalize the method to the class of linear time periodic coefficients DDEs with arbitrary period and multiple discrete and distributed delays. The scheme is shown to have spectral accuracy by means of several numerical examples
On discretizing the semigroup of solution operators for linear time invariant - time delay systems
in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(-\u3c4,0;C) and generalized Fourier projection when the state space is \u3c7 = C
7 L2(-\u3c4,0;C). Full discretization details for constructing the approximation matrices are given. Moreover, concise, yet fundamental, convergence results are discussed, with particular attention to their similarities and differences as well as pros and cons with regards to solution approximation and asymptotic stability detection
Equations with infinite delay : Numerical bifurcation analysis via pseudospectral discretization
We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.Peer reviewe
Realt\ue0 e Modelli
Tra le varie proposte presenti nel libro, il laboratorio \u201cRealt\ue0 e modelli\u201d vuole evidenziare il ruolo chiave della matematica nella modellizzazione di fenomeni reali, puntando anche a sviluppare un\u2019attitudine sperimentale verso la disciplina. Presenta le attivit\ue0 svolte con gli studenti delle scuole superiori relative alla descrizione e allo studio qualitativo e numerico di alcuni semplici modelli matematici che si incontrano nella fisica e nella biologia e che sono descritti da equazioni differenziali ordinarie
A numerical method for the stability analysis of linear age-structured models with nonlocal diffusion
We numerically investigate the stability of linear age-structured population
models with nonlocal diffusion, which arise naturally in describing dynamics of
infectious diseases. Compared to Laplace diffusion, the analysis of models with
nonlocal diffusion is more challenging since the associated semigroups have no
regularizing properties in the spatial variable. Nevertheless, the asymptotic
stability of the null equilibrium is determined by the spectrum of the
infinitesimal generator associated to the semigroup. We propose to approximate
the leading part of this spectrum by first reformulating the problem via
integration of the age-state and then by discretizing the generator combining a
spectral projection in space with a pseudospectral collocation in age. A
rigorous convergence analysis is provided in the case of separable model
coefficients. Results are confirmed experimentally and numerical tests are
presented also for the more general instance.Comment: 23 pages, 11 figure
Computing the eigenvalues of realistic Daphnia models by pseudospectral methods
This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled tothe evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work