Phase portraits of (2;1) reversible vector fields of low codimension

TL;DR

The article classifies and diagrams the phase portraits and bifurcations of planar $\varphi$-reversible vector fields of type $(2;1)$ with a line of reversibility, using Teixeira’s codimension normal forms up to codimension two. It combines local analyses at finite and infinite equilibria, quasihomogeneous and directional blow-ups, and Poincaré compactification to build global bifurcation diagrams in the Poincaré disk for the families $X_{01}$ through $X_{25}$, with a focus on symmetric singularities near the origin. The main contributions include explicit normal forms for codim0–2, comprehensive phase portraits and bifurcation structures for the codimension-two families, and detailed connections to local models of symmetric singularities. These results extend the qualitative understanding of reversible planar systems and provide a robust framework for recognizing and predicting bifurcations in related reversible dynamics, complementing prior quadratic-case classifications.

Abstract

In this paper we study the phase portraits and bifurcation diagram of the symmetric singularities of codimensions zero, one and two of planar reversible vector fields having a line of reversibility.

Figures (30)

• Figure 1: Local phase portrait of $X_{12}$ at the origin with $\lambda=0$.
• Figure 2: Local phase portrait at the infinity of $X_{12}$.
• Figure 3: Local phase portrait of the regularized infinity of $X_{12}$.
• Figure 4: Local behavior of the phase portrait of $X_{12}$ for $\lambda<0$.
• Figure 5: Local behavior of the phase portrait of $X_{12}$ for $\lambda=0$.

Theorems & Definitions (49)

• Definition 1: Topological equivalence
• Definition 2: $k$-parameter families
• Theorem 1: Teixeira Teixeira
• Theorem A
• Definition 3
• Theorem B
• Remark 1
• Remark 2
• Proposition 1
• proof