Locating and Stabilizing Unstable Periodic Orbits
Miino, Yuu Tokyo University of Technology
Ito, Daisuke Gifu University
Ueta, Tetsushi Tokushima University Tokushima University Educator and Researcher Directory KAKEN Search Researchers
Kawakami, Hiroshi Tokushima University Tokushima University Educator and Researcher Directory
unstable periodic point
Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.
International Journal of Bifurcation and Chaos
This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License(https://creativecommons.org/licenses/by/4.0/) which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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