Faster Computation of Periodic Orbits

The Lindstedt-Poincare technique in perturbation theory can be turned into a quadratically convergent algorithm for computing periodic orbits of differential equations. Most of the computational work is done in solving linear systems of the form ż(τ) =A(τ)z+g(τ), A(τ)2Rd;d , z2Rd , g(τ)2Rd , where A(τ) and g(τ) are 2π periodic. The method given here for solving such linear systems, while not the best way to solve such linear systems in every situation, is accurate enough to preserve the smooth quadratic convergence of the Lindstedt-Poincare algorithm. It uses the FFT to reduce the cost of solution from at least O(n2 logn) to O(n logn), where n is the length of the Fourier series used to represent A(τ) and g(τ). We explain the ruinous effect of untreated high frequency errors and their proper handling, and derive a multiple shooting version of the Lindstedt-Poincare algorithm. We compute an example showing three equal masses held together by gravity weaving around each other so that their periodic orbits form a braid.