Functions from $R^2$ to $R^2$: a study in nonlinearity

TL;DR

The paper tackles the geometry of maps $F: {\mathbb{R}}^2 \to {\mathbb{R}}^2$ by developing a framework that counts and computes preimages using local singularity theory and numerical continuation. It integrates Whitney's classification of critical points, rotation numbers, covering maps, and the continuation method, introducing the flower construction to organize local-to-global information and enabling practical preimage computation. Applying this to the test map $F_0$ reveals a rich structure of critical curves, preimage counts (including $F_0(z)=0$ having nine solutions), and dynamic birth of preimages as paths cross the critical set. The work demonstrates how topological invariants and continuation techniques combine to yield robust, computable insights into nonlinear plane-to-plane mappings with potential extensions to bounded domains and higher-dimensional targets.

Abstract

We study the geometry of functions from the plane to the plane. For a large special class we are able to count preimages and compute them. Both numerical and theoretical aspects are discussed. Some of the tools used are Whitney's classification of critical points, rotation numbers, covering maps and continuation methods.

Figures (15)

• Figure 1: Three different regimes (radii $0.1$, $1$ and $10$)
• Figure 2: Interpolating regimes: radii $0.2$, $0.3$, $0.7$, $1.5$ and $2$
• Figure 3: Local behavior near a fold
• Figure 4: Local behavior near a cusp
• Figure 5: A simple predictor-corrector method

Theorems & Definitions (15)

• Theorem 1: Whitney,W
• Proposition 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Proposition 8
• Theorem 9
• Theorem 10: Hopf, H