Direct computation of period doubling bifurcation points of large-scale systems of ODEs using a Newton-Picard method

Periodic solutions of certain large-scale systems of ODEs can be computed efficiently using a hybrid Newton-Picard scheme, especially in a continuation context. In this paper we describe and analyse how this approach can be extended to the direct computation of period doubling bifurcation points. The Newton-Picard scheme is based on shooting and a splitting of the state space in a low-dimensional subspace corresponding to the weakly table and unstable modes and its orthogonal complement. The method avoids the computation of the full monodromy matrix, which is present in the determining system for period doubling bifurcation points. Test results are presented that demonstrate the numerical properties.