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

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 M bifurcation software C MM, 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 J d for dth derivatives with SD. For small J, SD is still faster than AD.

Subharmonics and chaos in simple periodically-forced biomolecular models

“…To clarify our computational approach, we have to note that the default setup of MatcontM assumes (implicitly) that the map Φ (z, α) is given by explicit analytic expressions (formulas), since it uses algorithmic differentiation to compute Poincaré normal form coefficients [45]. Because our map Φ (z, α) is defined implicitly through a number of numerical integration steps, we disabled the algorithmic (also called automatic or symbolic) differentiation (adtayl) feature by setting 'AutDerivative' = 0.…”