Jaume Llibre And Claudia Valls
Phase Portraits of The Complex Riccati Equation With Constant Coefficients
Dynamic Systems and Applications 32 (2023) 189-198

https://doi.org/10.46719/dsa2023.32.11

ABSTRACT.
In this paper we characterize the phase portrait of the complex Riccati quadratic polynomial differential systems

dz/dt = ż = a(z-b)(z-c),

with z ϵ C, a,b,c ϵ C with a ≠ 0 and t ϵ R. Taking z=x+iy, and writing the Riccati equation as the differential system ( ẋ, ẏ ) in the plane, we give the complete description of their phase portraits in the Poincare disk (i.e. in the compactification of R² adding the circle S¹ of the infinity) modulo topological equivalence.

AMS (MOS) Subject Classification. Primary 34A05. Secondary 34C05, 37C10.

Key Words and Phrases.Complex Riccati equation, Poincar\’e compactification, dynamics at infinity.