Alternate locations of equilibrium points and poles in complex rational differential equations

Koh Katagata

doi:10.12988/imf.2014.410170

Abstract

We study configurations of simple equilibrium points of first order complex differential equations consisting of the iteration of rational functions. Rational functions which we deal with have the unit circle or the extended real line as Julia sets. Properties of Julia sets and the EulerJacobi formula lead to alternate locations of equilibrium points and poles of the complex differential equations. Mathematics Subject Classification: 37C10, 32A10, 37F10

Highlights

Let D be a domain in C and let f : D → C be a holomorphic function
We identify the differential equation (DE) with the system of differential equations x = u(x, y), y = v(x, y) in R2
Equilibrium points are categorized as stable nodes, unstable nodes, centers, stable foci, unstable foci and saddles

Summary

Introduction

Let D be a domain in C and let f : D → C be a holomorphic function. We consider the first order differential equation dz z ≡ = f (z), dt (DE). Let ζ ∈ D be an equilibrium point of the differential equation (DE). Configurations of equilibrium points of the complex differential equation (DE : c ; n) are as follows. Considering properties of the complex differential equation (DE : c ; n) and the Julia set of the rational function fc, we can obtain the analogical results to Theorem 1.2. (b) All equilibrium points of the differential equation (DE : a ; θ ; n) are symmetric with respect to the real axis. (c) Every equilibrium points of (DE : a ; θ ; n) on the real axis are unstable nodes. (d) Equilibrium points of (DE : a ; θ ; n) on the real axis and poles of A∗n are located alternately

Dynamics of rational functions A and B

Configurations of equilibrium points