Publication: Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation
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Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation

- Article in a journal -
 

Author(s)
J. D. Pryce , Khoshsiar Ghaziani , R. , De Witte , V. , W. Govaerts

Published in
Mathematics and Computers in Simulation

Year
2010

Publisher
Elsevier Science Publishers B. V.

Abstract
As an alternative to symbolic differentiation (SD) and finite differences (FD) for computing partial derivatives, we have implemented algorithmic differentiation (ad) techniques into the Matlab bifurcation software Cl_MatcontM, http://sourceforge.net/projects/matcont, where we need to compute derivatives of an iterated map, with respect to state variables. We use derivatives up to the fifth order, of the iteration of a map to arbitrary order. The multilinear forms are needed to compute the normal form coefficients of codimension-1 and -2 bifurcation points. Methods based on finite differences are inaccurate for such computations. Computation of the normal form coefficients confirms that ad is as accurate as SD. Moreover, elapsed time in computations using ad grows linearly with the iteration number J, but more like Jd for d th derivatives with SD. For small J, SD is still faster than ad.

AD Tools
CL_MatContM

AD Theory and Techniques
Taylor Arithmetic

BibTeX
@ARTICLE{
         Pryce2010Con,
       author = "Pryce, J. D. and Khoshsiar Ghaziani, R. and De Witte, V. and Govaerts, W.",
       title = "Computation of normal form coefficients of cycle bifurcations of maps by algorithmic
         differentiation",
       journal = "Mathematics and Computers in Simulation",
       issue_date = "September, 2010",
       volume = "81",
       number = "1",
       month = "sep",
       year = "2010",
       issn = "0378-4754",
       pages = "109--119",
       numpages = "11",
       url = "http://dx.doi.org/10.1016/j.matcom.2010.07.014",
       doi = "10.1016/j.matcom.2010.07.014",
       acmid = "1857366",
       publisher = "Elsevier Science Publishers B. V.",
       address = "Amsterdam, The Netherlands, The Netherlands",
       keywords = "Bifurcation, Iterated map, Matlab, Multilinear form, Taylor series",
       abstract = "As an alternative to symbolic differentiation (SD) and finite differences (FD) for
         computing partial derivatives, we have implemented algorithmic differentiation (AD) techniques into
         the Matlab bifurcation software Cl_MatcontM, http://sourceforge.net/projects/matcont, where we need
         to compute derivatives of an iterated map, with respect to state variables. We use derivatives up to
         the fifth order, of the iteration of a map to arbitrary order. The multilinear forms are needed to
         compute the normal form coefficients of codimension-1 and -2 bifurcation points. Methods based on
         finite differences are inaccurate for such computations. Computation of the normal form coefficients
         confirms that AD is as accurate as SD. Moreover, elapsed time in computations using AD grows
         linearly with the iteration number J, but more like Jd for d th derivatives with SD. For small J, SD
         is still faster than AD.",
       ad_tools = "CL_MatContM",
       ad_theotech = "Taylor Arithmetic"
}


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