Detecting Hidden Chaotic Regions and Complex Dynamics in the Self-Exciting Homopolar Disc Dynamo

Zhouchao Wei
School of Mathematics and Physics,
China University of Geosciences,
Wuhan 430074, P. R. China
Guangxi Colleges and Universities Key Laboratory
of Complex System Optimization and Big Data Processing,
Yulin Normal University, Yulin 537000, P. R. China
Mathematical Institute, University of Oxford,
Oxford, OX2 6GG, England
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, P. R. China
Zhouchao.Wei@maths.ox.ac.uk
weizhouchao@163.com

Irene Moroz
Mathematical Institute, University of Oxford,
Oxford, OX2 6GG, England

Julien Clinton Sprott
Department of Physics, University of Wisconsin,
Madison, WI 53706, USA

Zhen Wang
Department of Applied Sciences,
Xijing University, Xi’an 710123, P. R. China

Wei Zhang∗
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, P. R. China
sandyzhang0@yahoo.com

Received August 12, 2016; Revised October 12, 2016

ABSTRACT

In 1979, Moffatt pointed out that the conventional treatment of the simplest self-exciting homopolar disc dynamo has inconsistencies because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo. Here we return to the simple dynamo system proposed by Moffatt, and demonstrate previously unknown hidden chaotic attractors. Then we study multistability and coexistence of three types of attractors in the autonomous dynamo system in three dimensions: equilibrium points, limit cycles and hidden chaotic attractors. In addition, the existence of two homoclinic orbits is proved rigorously by the generalized Melnikov method. Finally, by using Poincar´e compactification of polynomial vector fields in three dimensions, the dynamics near infinity of singularities is obtained.

Ref: Z. Wei, I. Moroz, J. C. Sprott, Z. Wang, and W. Zhang, International Journal of Bifurcation and Chaos 27, 1730008 (2017)