Numerical computation of normal form coefficients of ODEs in MATLAB

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