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Abstract

This article continues earlier work of the author on non-linear systems of ordinary differential equations, published in Asymptotic Analysis 15 (1997), MR no .98g:34015b. There, a completely formal theory was presented, while here we are concerned with a semi-formal approach: Solutions of non-linear systems of ordinary meromorphic differential equations are represented as, in general divergent, power series in several free parameters. The coefficients, aside from an exponential polynomial, a general power and integer powers of the logarithm, contain holomorphic functions that are the multi-sums of formal power series. In J. Écalle's terminology such a semi-formal solution may be regarded as a transseries. In the author's opinion, however, they are best understood as power series in several variables. In this setting, we shall define and investigate the non-linear analogues of normal solutions, Stokes multipliers, and central connection coefficients, well known in the linear case. Moreover, we shall briefly address the question of convergence of the semi-formal series occurring. In particular, we wish to point out that in the cases when the series, due to the small denominator phenomenon, fails to converge, it is natural to be content with what shall be called partial convergence of the series, meaning that some of the variables are set equal to $0$, leaving a power series in fewer variables that then converges.