On diagrammatic technique for nonlinear dynamical systems

TL;DR

This paper develops a diagrammatic technique to analyze phase flows generated by polynomial vector fields in complex and real spaces, providing series expressions for evolution operators and examining their convergence and behavior near fixed points.

Contribution

It introduces a novel diagrammatic method to represent and analyze nonlinear dynamical systems with polynomial vector fields, including convergence estimates and fixed point behavior.

Findings

Series expressions for evolution operators using tree diagrams.

Convergence radius estimates in specific cases.

Analysis of phase flow behavior near fixed points.

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.