Year: 2025
Type: article
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 $$\begin{aligned} {\dot{w}}=P(w)/Q(w)\,, \qquad \qquad (*) \end{aligned}$$ for complex polynomials P , Q . We impose mild generic nondegeneracy conditions, including simplicity of poles and hyperbolicity of zeros. Generically, the real-time dynamics then 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 (*) on $$\widehat{{\mathbb {C}}}$$ 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 provides combinatorial counts for the number of global phase portraits, which only depend on the degrees of P and Q , respectively.
Author: Bernold Fiedler
Institution: Freie Universität Berlin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: Freie Universität Berlin
OpenAlex ID: W4416126266