Algorithms for Computing Periodic Orbits
This page describes multiple shooting algorithms for computing periodic
orbits that uses automatic differentiation. The method achieves accuracy
comparable to the floating point round-off of IEEE-754 double precision
arithmetic. The algorithms utilize three strategies that contribute
to their accuracy:
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Automatic differentiation is employed to calculate high order derivatives
of a vector field and its trajectories.
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The methods emphasize compact parametrizations of function spaces that
include high order approximations to the periodic orbits.
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A posteriori error estimates of the accuracy of numerically computed periodic
orbits are used for mesh adaptation.
The methods have additional attractive geometric features from both theoretical
and geometric perspectives. They utilize directly the geometric objects
that are prominent in the theory. These objects can be readily examined
and manipulated, and they can be used adaptively to enable the algorithms
to respond to changes in the geometry of a periodic orbit during continuation.
The algorithms give dense output, representations of approximate
periodic orbits at all points rather than just at mesh points of a discretization.
Constraints are readily imposed upon mesh points, enabling the accurate
computation of periodic orbits of piecewise analytic vector fields.
A detailed account of the mathematics underlying the algorithms is available
as a postscript manuscript: Computing Periodic
Orbits and their Bifurcations with Automatic Differentiation
that has been submitted for publication to the SIAM Journal on Scientific
Computing. The methods have been implemented in MATLAB, using a modified
version of the automatic differenentiation code ADOLC as an engine for
computing derivatives. is a gzipped tar file of
the computer codes with minimal documentation. These programs require MATLAB
and have been tested on a Sun Workstation with the Solaris 7 operating system.
The figure below shows a family of periodic orbits that are called canards.
