Periodic orbits of the ABC flow with $A=B=C=1$

TL;DR

This paper proves the existence of specific periodic orbits in the ABC flow with equal parameters, demonstrating implications for turbulent combustion models and flame speed enhancement.

Contribution

It establishes the existence of certain periodic orbits in the ABC flow with A=B=C=1, linking dynamical systems to combustion theory.

Findings

Existence of periodic orbits with specific rotation vectors

Implication for maximal flame speed enhancement in turbulent combustion

Connection between flow dynamics and combustion modeling

Abstract

In this paper, we prove that the ODE system $$ \begin{align*} \dot x &=\sin z+\cos y\\ \dot y &= \sin x+\cos z\\ \dot z &=\sin y + \cos x, \end{align*} $$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters $A=B=C=1$, has periodic orbits on $(2\pi\mathbb T)^3$ with rotation vectors parallel to $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $\mathbb R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.