BibTeX
@ARTICLE{
         Witte2011Nco,
       author = "Virginie De Witte and Willy Govaerts",
       title = "Numerical computation of normal form coefficients of bifurcations of {ODE}s in
         {MATLAB}",
       journal = "Discrete and Continous Dynamical Systems Supplements",
       year = "2011",
       pages = "362--372",
       doi = "10.3934/proc.2011.2011.362",
       abstract = "Normal form coefficients of codim-1 and codim-2 bifurcations of equilibria of ODEs
         are important since their sign and size determine the bifurcation scenario near the bifurcation
         points. Multilinear forms with derivatives up to the fifth order are needed in these coefficients.
         So far, in the Matlab bifurcation software MatCont for ODEs, these derivatives are computed either
         by finite differences or by symbolic differentiation. However, both approaches have disadvantages.
         Finite differences do not usually have the required accuracy and for symbolic differentiation the
         Matlab Symbolic Toolbox is needed. Automatic differentiation is an alternative since this technique
         is as accurate as symbolic derivatives and no extra software is needed. In this paper, we discuss
         the pros and cons of these three kinds of differentiation in a specific context by the use of
         several examples.",
       keywords = "Automatic differentiation, symbolic derivatives, MatCont, directional derivatives",
       ad_tools = "MatCont",
       ad_theotech = "Taylor Arithmetic"
}