Computing periodic orbits and their bifurcations with automatic differentiation

This paper formulates several algorithms for the direct computation of peri odic orbits as solutions of boundary value problems. The algorithms emphasi ze the use of coarse meshes and high orders of accuracy. Convergence theore ms are given in the limit of increasing order with a fixed mesh. The algori thms are implemented with the use of MATLAB and ADOL-C, a software package for automatic differentiation. Automatic differentiation enables accurate c omputation of high-order derivatives of functions without the truncation er rors inherent in finite difference calculations. We embed the algorithms in a continuation framework and extend them to compute saddle-node bifurcatio ns of periodic orbits directly. We present data from numerical studies of f our test problems, making some comparisons with other methods for computing periodic orbits. These results demonstrate that high-order methods based u pon automatic differentiation are capable of high precision with small mesh es.