The well-known matrix exponentiation for linear first order systems of ODE is generalised to polynomial vector fields. The substitution process of our Exponential Formula is closely related to the usual iteration of polynomial mappings.
1997 Academic Press
The n-dimensional linear autonomous system of differential equations x* =Mx, x # R n , M # R n_n has the flow .(t)=e Mt := i=0 (M i รi !) t i defined for all t # R, where M i is the i th power of the coefficient matrix. Application to an initial value x 0 # R n gives the global solution .(t, x 0 )=e Mt x 0 of the IVP x* =Mx, x(0)=x 0 . The phaseportrait of such a linear system is completely understood, when one knows it in some neighbourhood of the origin in fact this is true for all homogeneous systems. Furthermore in the linear case the global flow is already determined by the infinitesimal, i.e. algebraic, structure of the system and explicit solutions can be calculated using the Jordan normal form of M. All this is very well known (cf. [HS]).
In this paper we describe a simple straightforward generalisation of the matrix exponentiation, which gives the local flow for polynomial vector fields x* =p(x), x # R n , p # (R[x]) n , p of degree m 2 in the form exp(t+)= i=0 (t i รi !) + i . In this formula + is a multilinear mapping corresponding to the homogenisation of p, and the + i are sums of suitably composed multilinear mappings, which generalise the matrix powers M i . The structure of this composition can be represented in a natural way by graphs, namely by forests of certain rooted trees (see [W1]). Application of exp(t+) to an initial value x 0 # R n then gives the analytic power series solution of the (homogenised form of the) polynomial IVP'' x\* =p(x), x(0)=x 0 . We prove these facts in Section 1 using a fundamental recursion formula for the + i . Moreover this section includes a proof of the 1-parameter-group property of exp(t+) and remarks on the history'' of the Exponential Formula and on homogeneous polynomial vector fields. Two further recursion formulas for the coefficients of exp(t+) x 0 , the article no. AI971638 190