Quadratic differential equations in the complex domain I

Abstract

By complexifying all of the variables of an ordinary real quadratic vector differential equation to get a differential equation over C \mathbb {C} , it is shown that the solution to the complex differential equation can be uniquely defined on an open star-shaped subset of C \mathbb {C} , dependent on the initial point, containing the maximum interval of existence of the real differential equation. Complex conjugation is shown to commute with solving the differential equation on this complex domain, and well-known algebraic properties of the solutions to the real differential equation are generalized to the equation over C \mathbb {C} .