Periodic orbits of hybrid systems and parameter estimation via AD Periodic processes are ubiquitous in biological systems, yet modeling these processes with high fidelity as periodic orbits of dynamical systems is challenging. Moreover, mathematical models of biological processes frequently contain many poorly-known parameters. This paper describes techniques for computing periodic orbits in systems of hybrid differential-algebraic equations and parameter estimation methods for fitting these orbits to data. These techniques make extensive use of automatic differentiation to evaluate derivatives accurately and efficiently for time integration, parameter sensitivities, root finding and optimization. The resulting algorithms allow periodic orbits to be computed to high accuracy using coarse discretizations. Derivative computations are carried out using a new automatic differentiation package called ADMC++ that provides derivatives and Taylor series coefficients of matrix-valued functions written in the MATLAB programming language. The algorithms are applied to a periodic orbit problem in rigid-body dynamics and a parameter estimation problem in neural oscillations.
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- Phipps, Eric; Casey, Richard; Guckenheimer, John: Periodic orbits of hybrid systems and parameter estimation via AD (2006)