Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems
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RESEARCH PAPER
Harmonic balance‑based approach for optimal time delay to control unstable periodic orbits of chaotic systems Y. M. Chen1 · Q. X. Liu1 · J. K. Liu1 Received: 29 February 2020 / Revised: 6 May 2020 / Accepted: 28 May 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract As a classical technique for chaos suppression, the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits (UPOs) embedded in chaotic systems. A critical issue for achieving high controlling precision is to search for an appropriate time delay. This paper proposes a simple yet effective approach, based on incremental harmonic balance method, to determine the optimal time delay in the delayed feedback controller. The time delay is adjusted within the iterative scheme provided by the proposed method, and finally converges to the period of the target UPO. As long as the optimal time delay is fixed, moreover, the attained solution makes it quite convenient to analyze its stability according to the Floquet theory, which further provides the effective interval of the feedback gain. Keywords Chaos control · Unstable periodic orbit · Delayed feedback · Optimal time delay · Incremental harmonic balance method
1 Introduction Due to the promising potential applications in physics, chemical reactor, control theory, biological networks, artificial neural networks and telecommunications technology, chaos suppression has become a fast-developing interdisciplinary research field in the last two decades [1–5]. It has received a great amount of research interest and applications from various disciplines such as physics [6], physiology [7], system science [8, 9], economy [10] and ecology [11], to mention a few. Generally speaking, chaos controlling methods can be categorized as “feedback control” such as the renowned Pyragas method [12], and “nonfeedback control” or “feedforward control” such as the famous Ott-Grebogi-Yorke (OGY) method [13] (based on linearization of the Poincare map) [14]. Since Ott et al. [13] initiated the OGY method, the preferable strategy for chaos suppression is usually designed to stabilize unstable periodic orbit (UPO) that * Y. M. Chen [email protected] 1
Department of Mechanics, Sun Yat-sen University, Guangzhou 510275, China
embedded in the considered chaotic systems [15–20]. With such simplicity and efficiency, the time delayed feedback controller has also been widely employed for stabilizing UPOs [21–24]. Though without strict requirement for much prioriknown knowledge about the target UPOs, in real practice much attention has to be put on selecting the time delay as well as the gain coefficient of the feedback. For this issue, a variety of valuable modifications have been made to achieve adaptive tuning of certain critical parameters. For instance, Socolar et al. [25] suggested an extended time delayed feedback controller by considering multiple time delays in the suppre
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