Periodic Solutions of Holomorphic Differential Equations

Abstract

This chapter describes the periodic solutions of holomorphic differential equations. It presents an assumption in which M is a complex manifold and F is the collection of holomorphic time-dependent vector fields of (fixed) period ω. Therefore, ƒ ∈ ɛ when f maps M x ℝ continuously into TM, the tangent bundle of M, in such a way that (1) ƒ (x, t + ω) = ƒ (x,t), (2) f is holomorphic in x, and (3) f(·, t) is a section of TM for each t. ɛ is given the topology of uniform convergence on compact sets of M × ℝ. The orbits of ƒ are given locally by the solutions of an ordinary differential equation ɛ = ƒ* (t, z).