非自治系統之複雜動態行為及混沌控制; Complex Dynamics and Chaos Control of Nonautonomous systems

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Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/1798

Title:	非自治系統之複雜動態行為及混沌控制;Complex Dynamics and Chaos Control of Nonautonomous systems
Authors:	蕭永嘉;Yung-Chia Hsiao
Contributors:	機械工程研究所
Keywords:	非自治系統;雙共同維度;分歧;穩定週期解;沌控制;nonautonomous system;codimension-two
Date:	2001-07-19
Issue Date:	2009-09-21 11:32:04 (UTC+8)
Publisher:	國立中央大學圖書館
Abstract:	在本論文中首先研究一個不對稱非自治系統的動態行為。此系統的動態行為顯示雙共同維度分歧為雙不穩定週期解合併所組成之鞍點?節點分歧的發生機制。此雙共同維度分歧由鞍點?節點分歧與週期加倍分歧交會所組成。另外由此系統的動態行為發現非自治系統的主要響應與第二響應會互相融合。此外，這個不對稱非自治系統有混沌現象存在。許多非線性系統並不希望有混沌現象產生。一般利用混沌控制來消除系統中的混沌現象。設計混沌控制器前必須先標定某個存於混沌軌跡中的不穩定週期解。本論文以尤拉法來推導非自治系統之全域龐加萊映射的近似方程式，並利用此映射方程式來標定存在於混沌軌跡中的不穩定週期解。傳統的混沌控制為區域式控制器。在啟動混沌控制器前必須有一段極冗長的等待時間。本論文以推導出之全域龐加萊映射來估算被混沌控制器穩定化之週期解的收斂區。利用此收斂區可有效減少混沌控制的等待時間。此外，為完全消除此等待時間，本論文利用全域龐加萊映射設計一個全域式混沌控制器。此控制器移除除選定的不穩定週期解外其他的不穩定週期解，並將此唯一的週期解漸進穩定化。如此全域式混沌控制器可在混沌現象出現時便將之消除，完全不需要等待時間。 A saddle-node bifurcation with the coalescence of a stable periodic orbit and an unstable periodic orbit is a common phenomenon in nonlinear systems. This study investigates the mechanism of producing another saddle-node bifurcation with the coalescence of two unstable periodic orbits. The saddle-node bifurcation results from a codimension-two bifurcation that a period doubling bifurcation line tangentially intersects a saddle-node bifurcation line in a parameter plane. Furthermore, this thesis investigates a coalescence of the primary responses and the secondary responses in the asymmetric nonautonomous system. A subharmonic orbit that bifurcates from the primary responses coalesces with a subharmonic orbit of the secondary responses via a saddle-node bifurcation. In addition, the output of the nonautonomous system is chaotic in a specific parameter range. The chaotic motion is generally undesirable to a nonautonomous system. To control a chaotic motion to an unstable periodic orbit embedded in a chaotic trajectory, detection of the unstable periodic orbits from a chaotic time series is necessary to implement the control. This thesis presents a simple approach that detects unstable periodic orbits embedded in a chaotic motion of an unknown nonautonomous system with noisy perturbation. An identification technique is developed to obtain the model of the unknown system. The nonautonomous system is approximated by a difference system and then a global Poincaré map function is derived from the difference system. The unstable periodic orbits can be detected via the map function. The proposed method is both accurate and feasible as demonstrated by two chaotic nonautonomous systems. Many local controls of chaos were studied to suppress chaotic motions. However, there is tedious waiting time before activating the controllers. This thesis develops a strategy of controlling chaos with a region of attraction of a stabilized UPO. The strategy is activated when chaotic trajectories get into the region of attraction. The region of attraction is estimated via the approximate global Poincaré map function. The proposed strategy considerably reduces a lot of the waiting time of controlling chaos. To suppress the waiting time completely, this thesis develops a global control of chaos. The proposed global controller, who does not require waiting time in activating the controller, can be rapidly started to stabilize the targeted UPO. The global controller makes the all unstable periodic orbits vanish except a targeted unstable periodic orbit. Furthermore, a Lyapunov’s direct method is applied to confirm that the global controller can asymptotically stabilize the unique periodic orbit. Simulation results demonstrate that the global controller successfully regularizes a chaotic motion even if the chaotic trajectory is far from the targeted periodic orbit.
Appears in Collections:	[Graduate Institute of Mechanical Engineering] Electronic Thesis & Dissertation

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