Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations

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

“…In the case of PDEs, the eigenfunctions are still exponential in time, but the order of regularity with respect to the physiological variables depends on the regularity of the model parameters, which therefore affects the order of convergence of the approximation [43,Theorem 3.2]. A similar behavior has been shown for the approximation of R 0 for structured epidemic models [5,8,11], and in the approximation of the solution operators [12,13,15].…”

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

“…The version based on polynomials on a single piece was presented for DDEs in [16,17], for REs in [11] and for coupled equations in [12], along with the proofs of the well-posedness of ( 7) and of the convergence of the approximated eigenvalues. 8 In this work we present an implementation of the version based on continuous piecewise polynomials.…”

“…We can find some bifurcations by using MATLAB's fzero on a function testing for one multiplier crossing the unit circle. 12 We first create the test function as a file test _ quadraticre _ bif.m with the following content. We can then find two bifurcations (a Hopf bifurcation at γ = 3.570796208333382 and a period doubling bifurcation at γ = 4.325285374879225) with these instructions.…”

“…Extensions of [16] have been recently proposed for renewal equations (REs) [11] and systems of REs and DDEs [12]. In fact, as far as we know, these are the first available methods for REs.…”

Piecewise discretization of monodromy operators of delay equations on adapted meshes

“…Accordingly, we highlight that when the linear system comes from linearizing a nonlinear problem around a numerically computed periodic solution, the resulting coefficients are in general only continuous, even for smooth problems, being the approximated solution a continuous piecewise polynomial. This may deteriorate the convergence of the computated multipliers, as it happens, e.g., for the pseudospectral collocation methods for DDEs [13], for REs [9] and for coupled REs and DDEs [10]. All these methods construct a matrix discretizing the monodromy operator by using a single polynomial on the whole domain interval.…”