Dynamics of Delay Equations, Theory and Applications - Abstract
Scarabel, Francesca
The increasing importance of delay equations in mathematical models urges the development of methods for the numerical investigation of their dynamical properties. Today, some software is available for delay differential equations with discrete delays and for the equilibrium continuation of specific delay models, but no tailor-made tools exist for the numerical bifurcation analysis of renewal equations or differential equations with distributed delay. The pseudospectral discretization of a nonlinear delay equation allows to approximate the original infinite-dimensional system by a low-dimensional system of ordinary differential equations, which can be studied by means of well-established numerical tools. Thanks to the special structure of the problem, the approximating system can be easily written from the right-hand side of the original equation, potentially providing a systematic procedure for the numerical bifurcation analysis of delay equations. By means of some examples, we show how the approach can be used to efficiently obtain reliable bifurcation diagrams of renewal equations and systems of coupled renewal and delay differential equations.