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

Abstract

In this paper, we prove that the ODE system $$\begin{array}{rl}\dot{x}& =\sin z+\cos y\\ \dot{y}& =\sin x+\cos z\\ \dot{z}& =\sin y+\cos x,\end{array}$$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters $A=B=C=1$, has periodic orbits on $(2\ddot{I}€\mathbb{T}{)}^3$ 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.

9 pages