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ID 115874
Title Alternative
Locating and Stabilizing Unstable Periodic Orbits
Author
Miino, Yuu Tokyo University of Technology
Ito, Daisuke Gifu University
Keywords
Chaos
horseshoe map
symbolic dynamics
unstable periodic point
numerical computation
controlling chaos
Content Type
Journal Article
Description
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.
Journal Title
International Journal of Bifurcation and Chaos
ISSN
02181274
17936551
NCID
AA10810319
Publisher
World Scientific
Volume
31
Issue
4
Start Page
2150110
Published Date
2021-03-30
Rights
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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language
eng
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departments
Center for Administration of Information Technology
Science and Technology