Investigation Of Bifurcation Control For Nonlinear Dynamical Systems With Time Delay Feedback

As a leading subject and new interests of nonlinear researches, bifurcation control and time delay dynamics come up with great challenge. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable behaviors. Through a complete summary and examination of the history and the actuality of the bifurcation control research, a systematic investigation into the fundamental theory and application of the bifurcation control of a differential nonlinear system has been studied based on the nonlinear vibration theory, the nonlinear dynamics theory, the bifurcation theory and the time delay dynamics theory, and the time delay feedback controller has been designed. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of bifurcation control. The main contents in this paper can be stated as follows:Chapter one outlines some study methods, nonlinear theory and recent advances on bifurcation control first, and introduces the aim, contents and innovations of the study in this paper.Chapter two introduces some basic conceptions on dynamics. The necessary and sufficient conditions of genesis for three elementary static bifurcations, including saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and transfer conditions between each other are introduced. Then some design and analysis methods on bifurcation controller and time delay dynamics are presented.In chapter three, a controller with two linear time delays-----displacement andvelocity, is designed to the forced Duffing system with quadratic and cubic nonlinearities, Studies cover primary resonance, subresonance and superresonance. The critical conditions of bifurcation are obtained based on the study. The saddle-node bifurcation is eliminated as well as the resonance amplitude is reduced; the relationship between steady response amplitude and controller parameters is obtained and design method of this kind time delay controller is provided.In chapter four, a linear controller with time delay displacement and time delay velocity is designed to van der Pol-Duffing system, and principle parametric resonance is studied. Using the perturbation method, the average equations and bifurcation equation are obtained. Studying the obtained equations, the effects of main parameters to the bifurcation are found, so the control strategy is advanced. Thedesirable controller is designed and the bifurcation of this system is controlled well.As an engineering case, a post-bulked dynamics bifurcation of a beam subjected to harmonic axial excitation is studied. Using the time delay velocity feedback controller, the critical force control is achieved. The study indicates this controller can effectively eliminate super-critical bifurcation or change the position of sub-critical bifurcation.In the chapter five, van der Pol-Duffing system with the controller including linear time delay items and nonlinear items at the same time is investigated. Hopf bifurcation for the static steady case is studied and time delay parameters' effects on the condition of this bifurcation are obtained. For the steady periodic resonance case, the relationship between limit cycle amplitude and time delay parameters is obtained, so the aim of amplitude control of limit cycle with time delays is achieved.In the chapter six, a nonlinear beam under moving loads is investigated. The dynamics differential equation with time delays is obtained and the dynamics behaviors of this system are analyzed. Particularly, a kind of nonlinear time delay controller is introduced, which can obtain the results that cannot be obtained with the linear time delay controller.In this paper, some innovative thinking is that using bifurcation control theory investigates the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear theory. The incorporates are as follows:1. The time delay feedback control is used to investigate nonlinear dynamic systems control, especially for the bifurcation control. Some kind of time delay feedback controllers are designed and studied. The systems' dynamic behaviors are optimized and some control strategies for the various systems are obtained.2. Time delay controllers with multi time delays are new exposure in the interrelated field. In this paper, controller with two time delays, including displacement and velocity, is investigated. Using the designed controllers, the Duffing system with quadratic and cubic nonlinearities and van der Pol-Duffing system with parametric motivation are investigated. The results indicate these controllers can be well used to bifurcation control of primary resonance.3. Using linear time delay velocity controller, the flexible post-bulking beam under parametric excitation and harmonic motivation is investigated. The aims, including controlling critical force, eliminating super-critical bifurcation and changing the bifurcation position of sub-critical bifurcation, are achieved.4. A nonlinear beam under moving loads is studied. The dynamics differentialequation with time delays is obtained and the dynamics behaviors of this system are analyzed. A nonlinear time delay controller is obtained, which can content the bifurcation aims and meantime make the response amplitude invariably. This result is meaning for practical engineering.Through the present research work, the bifurcation control theory is enriched and developed. The studied have put forward the theoretical foundations and approaches for designing time delay feedback controllers for optimizing the dynamics behaviors. The bifurcation control is a new researching field that needs not only more deeper and wider theoretical studies but practical applications. Due to the widely existence of time delays and the superiority of time delay control, it can be anticipated that new results publications on this subject will be appeared, more and more researchers will pursue further in this stimulating and promising direction of the new research.