We describe efficient methods for the computation and continuation of periodic solutions of parabolic partial differential equations (PDEs) and delay differential equations (DDEs).
A space discretization of a PDE results in a large system of ODEs, which often exhibits only low-dimensional dynamics. Newton-Picard methods exploit this property, by computing periodic solutions using a combination of Newton-based shooting in the low-dimensional subspace where the solution is unstable or weakly stable and a Picard iteration in the orthogonal complement of that subspace. This approach greatly reduce the computational cost, when compared with classical single or multiple shooting. Good estimates for the dominant Floquet multipliers are obtained at no extra cost and can easily be refined.
The Newton-Picard framework can also be used to compute periodic solutions of delay differential equations with several delays. A shooting approach, together with a discretization of the initial function and of the Poincaré operator leads to a large nonlinear system, which can be solved efficiently by a Newton-Picard method, due to the spectral properties of the (discretized) Poincaré operator. Also stability information is available, since (approximations to) the dominant Floquet multipliers are computed.
Newton-Picard methods are especially efficient when they are used in a continuation procedure. The approach can be generalized to construct extended systems for the accurate computation of bifurcation points along branches of periodic solutions (e.g. period doubling points).
This is joint work with Kurt Lust, Koen Engelborghs and Tatyana Luzyanina.