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 w˙ = P(w)/Q(w) , (∗) 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 ∈ C , we classify the resulting global dynamics up to C0 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 C realize all finite multi-graphs on S2, 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.
