Complex Dynamics and Chaos Control of Nonautonomous systems

• Main Authors: Yung-Chia Hsiao, 蕭永嘉
• Other Authors: Pi-Cheng Tung
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/02354728497908557938

博士 === 國立中央大學 === 機械工程研究所 === 89 === 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.