Repeated Eigenvalues Imply Nodes? A Problem of Planar Differential Equations

TL;DR

Revisits Poincaré's planar-equilibrium criterion via Jacobian eigenvalues, noting that for a $C^1$ system the standard conclusions hold when $\lambda_1\neq\lambda_2$ but can fail when the eigenvalues coincide; strengthens the regularity to $C^{1,\alpha}$ to recover the criterion in the repeated-eigenvalue case. The paper outlines a proof strategy based on diagonalization and polar-coordinate analysis, with Grobman-Hartman theorem supporting the topological aspects. It provides an explicit $C^1$ counterexample showing a focus with $\lambda_1=\lambda_2$, and then proves that for $C^{1,\alpha}$ nonlinear planar systems the $\lambda_1=\lambda_2$ case yields a node (attracting or repelling) after considering both diagonalizable and non-diagonalizable Jacobians. The results clarify the regularity thresholds needed to extend linear-classification results to nonlinear planar systems and guide interpretation of phase portraits under repeated eigenvalues.

Abstract

Poincaré gave a criterion which determines the shape of equilibrium for planar differential equations. In his statement, he excluded the case of repeated eigenvalues. In fact, in such a case, we can give a $C^1$ counter-example to his assertion. In this note, we show that if we strengthen the condition to $C^{1,α}$ ($0<α<1$), his assertion becomes true even in case of repeated eigenvalues.

Theorems & Definitions (5)

• Theorem 1
• Example
• Theorem 2
• proof : Proof of Example
• proof : Proof of Theorem 2.