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

Jack Xin University of California at Irvine Yifeng Yu University of California at Irvine Andrej Zlato University of Wisconsin–Madison

Dynamical Systems mathscidoc:1608.11001

ArXiv, 2016.1
In this paper, we prove that the ODE system, $ \dot{x}=sinz+cosy$ $ \dot{y}=sinx+cosz$ $ \dot{z}=siny+cosx$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, has periodic orbits on $(2 \pi T)^3$ 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 $R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.
Arnold-Beltrami-Childress (ABC) flow, periodic orbits, G-equation model
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@inproceedings{jack2016periodic,
  title={Periodic orbits of the ABC flow with $A=B=C=1$},
  author={Jack Xin, Yifeng Yu, and Andrej Zlato},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818094045740432224},
  booktitle={ArXiv},
  year={2016},
}
Jack Xin, Yifeng Yu, and Andrej Zlato. Periodic orbits of the ABC flow with $A=B=C=1$. 2016. In ArXiv. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818094045740432224.
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