Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere

Abstract

Inspired by pioneering work of Kyûya Masuda in the 1980s, only much more recent PDE studies address global boundedness versus finite-time blow-up in complex time. The two phenomena are related by passage from real to purely imaginary time. As a most simplistic ODE example, we study scalar rational vector fields \begin{equation*} \label{*} \dot{w}=P(w)/Q(w), \tag{*} \end{equation*} for complex polynomials $P,Q$. We impose mild generic nondegeneracy conditions, including simplicity of poles and hyperbolicity of zeros. Generically, the real-time dynamics become gradient-like Morse. Poles play the role of hyperbolic saddle points. At poles, however, solutions may blow up in finite time. On the Riemann sphere $w\in\widehat{\mathbb{C}}$, we classify the resulting global dynamics up to $C^0$ orbit equivalence, in real time. This relies on a global description of the connection graph of blow-up orbits, from sources towards saddles/poles, in forward time. Time reversal identifies the dual graph of blow-down orbits. We show that the blow-up and blow-down graphs of (*) realize all finite multi-graphs on $\mathbb{S}^2$, equivalently. The purely polynomial case $Q=1$ realizes all planar trees, alias diagrams of non-intersecting circle chords. The anti-holomorphic cousin $P=1$ realizes all noncrossing trees with vertices restricted to circles. This classification provide combinatorial counts for the number of global phase portraits, which only depend on the degrees of $P$ and $Q$, respectively.