On diagrammatic technique for nonlinear dynamical systems

Abstract

In this paper we investigate phase flows over $\mathbb{C}^n$ and $\mathbb{R}^n$ generated by vector fields $V=\sum P^{i}\partial_i$ where $P^{i}$ are finite degree polynomials. With the convenient diagrammatic technique we get expressions for evolution operators $ev\{V|t\}: x(0)\mapsto x(t)$ through the series in powers of $x(0)$ and $t$, represented as sum over all trees of particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.