Piecewise discretization of monodromy operators of delay equations on adapted meshes

Abstract

<p style='text-indent:20px;'>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.</p>

Magnus integrators for linear and quasilinear delay differential equations

“…In this way we may add yet another tool to the arsenal already available to analyze all these issues, as shown e.g. in [10,[12][13][14]30] and references therein. Eq.…”

“…The goal of this technique is to derive systems of ordinary differential equations, analyze their stability and eventually obtain bifurcation diagrams in the space of parameters with the tools already available in the theory of dynamical systems. These include, in particular, the computation of Floquet multipliers [1,11], even for linear time-periodic DDEs with discontinuous coefficients [12,13]. This procedure has also been used to compute periodic solutions, in contrast with the classical approach based on piecewise orthogonal collocation [14].…”