Analysis of stationary points and their bifurcations in the ABC flow

TL;DR

This paper analytically characterizes stationary points in the ABC flow, including their existence, stability, and bifurcation behavior, providing explicit formulas and stability analysis.

Contribution

It offers the first analytical expressions for stationary points, their eigenvalues, and bifurcation behavior in the ABC flow.

Findings

Stationary points are saddle-node type.

Explicit eigenvalues and eigenvectors are derived.

Stationary points' behavior along bifurcation lines is described.

Abstract

Analytical expressions for coordinates of stationary points and conditions for their existence in the ABC flow are received. The type of the stationary points is shown analytically to be saddle-node. Exact expressions for eigenvalues and eigenvectors of the stability matrix are given. Behavior of the stationary points along the bifurcation lines is described.