Numerical bifurcation analysis of periodic solutions of partial differential equations

There is a growing interest in the study of periodic phenomena in largescale nonlinear dynamical systems Often the highdimensional system has only lowdimensional dynam ics We propose an approach that exploits this property in order to compute branches of periodic solutions of such systems eciently We call our approach the NewtonPicard method We rst study a family of techniques based on the single shooting method The linearized single shooting system is projected on the lowdimensional subspace of unstable and weakly stable modes and its orthogonal complement In both subspaces dierent numerical techniques are used In its simplest form our method is equivalent to time integration in the highdimensional subspace followed by Newtonbased single shooting in the lowdimensional subspace The convergence of several variants of the method is thoroughly analyzed and we exploit the analysis to construct variants that are nearly as robust as Newtonbased single shooting with a direct linear system solver As a side eect of the construction of the projectors we obtain good estimates for the dominant stabilitydetermining Floquet multipliers Hence the method is particularly suited for bifurcation analysis The approach is extended to multiple shooting The algorithm is shown to be almost as ecient as the singleshooting based methods However multiple shooting is a much more robust method We pay special attention to the accurate computation of the Floquet multipliers for highly unstable limit cycles We show that the NewtonPicard idea can also be applied to extended systems for the direct computation of bifurcation points using both single and multiple shooting based techniques i i