Solving nonlinear equations from higher order derivations in linear stages (Q752828)

This paper is concerned with effective algorithms to compute approximations to higher order derivations. Let \(R=C^{\infty}({\mathbb{R}}^ N)\) and let \(D_{\mu}=\partial /\partial x_{\mu}\), for \(1\leq \mu \leq N\). The data consists of M derivations \(F_ 1,...,F_ M\) of the form \(F_ j=\sum^{N}_{\mu =1}a_ j^{\mu}D_{\mu}\), where \(a_ j^{\mu}\in R\), \(1\leq j\leq M\). The problem is to find M derivations \(E_ 1,...,E_ M\), with polynomial coefficients with the property that the \(E_ j\) generate a nilpotent Lie algebra and such that trajectories of the nonlinear system \(\dot y(t)=\sum^{M}_{j=1}u_ j(t)F_ j(y(t))\), \(y(0)=0\) are close to trajectories of the system \(\dot x(t)=\sum^{M}_{j=1}u_ j(t)E_ j(x(t))\), \(x(0)=0\) in the sense that \(| y(t)-x(t)| \leq Ct^ r\), for small t. In general this estimate does not hold for \(r\geq 2.\) This paper gives an algorithm which computes the derivations \(E_ j\) for larger r, when possible. The basic idea is to introduce a data structure involving finite trees labeled by the derivations, and to impose a multiplication on trees so that the space of trees becomes an algebra homomorphic to the algebra of derivations generated by the \(F_ j\). A careful analysis of the algebra of trees then yields the algorithm.