A numerical method based on pseudospectral collocation is proposed to approximate the eigenvalues of evolution operators for linear renewal equations, which are retarded functional equations of Volterra type. Rigorous error and convergence analyses are provided, together with numerical tests. The outcome is an efficient and reliable tool which can be used, for instance, to study the local asymptotic stability of equilibria and periodic solutions of nonlinear autonomous renewal equations. Fundamental applications can be found in population dynamics, where renewal equations play a central role.

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.

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 of 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.