There is a growing interest in the study of periodic phenomena in large-scale nonlinear dynamical systems. We call our approach the Newton-Picard method. Out technique tries to combine the performance of straightforward time integration with the advantages of solving a nonlinear boundary value problem using Newton's method and a direct solver.
Time integration works well for very stable limit cycles.
Solving a boundary value problem is expensive, but works also for unstable limit cycles. Next we will explain the basic features of the Newton-Picard method for single shooting. The linearised system is solved by a combination of direct and iterative techniques. While computing the projectors, we also obtain good estimates for the dominant, stability-determining Floquet multipliers. We will present a framework which allows us to monitor and steer the convergence behaviour of the method.