Computing Periodic Orbits and their Bifurcations with Automatic Differentiation

This paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems. The algorithms emphasize the use of coarse meshes and high orders of accuracy. Convergence theorems are given in the limit of increasing order with a fixed mesh. The algorithms are implemented with the use of MATLAB and ADOL-C, a software package for automatic differentiation. Automatic differentiation enables accurate computation of high-order derivatives of functions without the truncation errors inherent in finite difference calculations. We embed the algorithms in a continuation framework and extend them to compute saddle-node bifurcations of periodic orbits directly. We present data from numerical studies of four test problems, making some comparisons with other methods for computing periodic orbits. These results demonstrate that high-order methods based upon automatic differentiation are capable of high precision with small meshes.

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