About existence of stationary points for the Arnold-Beltrami-Childress (ABC) flow

TL;DR

This paper investigates the existence of stationary points in the ABC-flow, a fundamental model in fluid dynamics and chaos theory, proving the existence of specific stationary points for certain parameter cases.

Contribution

It demonstrates the existence of stationary points in the ABC-flow for particular parameter configurations, advancing understanding of its dynamical structure.

Findings

Existence of one stationary point for A=B=1

Existence of one stationary point for C=1 with A^2+B^2=1

Potential for three stationary points depending on parameters

Abstract

The existence of stationary points for the dynamical system of ABC-flow is considered. The ABC-flow, a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible, inviscid fluid flows, is the prototype for the study of turbulence (it provides a simple example of dynamical chaos). But, nevertheless, between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points, the deterministic basis among the chaotic behaviour of ABC-flow dynamical system. It has been proved the existence of 1 point for two partial cases of parameters {A, B, C}: 1) A = B = 1; 2) C = 1 (A^2 + B^2 = 1). Moreover, dynamical system of ABC-flow allows 3 points of such a type, depending on the meanings of parameters {A, B, C}.