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

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