Bifurcation And Chaos Control Of Duffing Systems Under Time Delayed State Feedback Control

Bifurcation, chaos and the theory and methods of chaos control in nonlinear During systems are discussed and summarized in this paper. Beginning with analysis of the global bifurcations of the soft nonlinear During system, based on further research and synthesis of some problems concerning soft nonlinear Duffing system in preceding work, such as the global dynamics behavior analysis and estimate of the size of attracting regions are discussed. The characteristic and the relationship of each subharmonic bifurcation of the homoclinic orbits, the inner orbits and the outer orbits of the homoclinic orbits varying with the parameters are discussed. The routes from the symmetry-breaking bifurcation to the period-doubling bifurcation of hard nonlinear Duffing systems are verified and studied carefully. The diversity and complexity of this transformation are put forward and investigated, and the processes and the results are demonstrated or illustrated. The development foreground and some problems which to be solved of nonlinear During systems under time delayed state feedback control are discussed especially. In addition, the numerical methods and the nonlinear time series analysis are also studied, and three new nonlinear 3-D autonomous chaotic flows are presented, which is lower or higher fractal dimension flow, even almost-Hamiltonian chaotic flow, to validate the advantage and accuracy of two formulas for calculating Lyapunov dimensions.The main contributions of the paper are as follows:1. The development history of bifurcation and chaos theory is reviewed and the conceptions, definitions of chaos are summarized. And a minute illustration of chaos research progress and its application are given. The significance, the background and the foreground of time delayed nonlinear dynamics are expatiated especially.2. The significance of studying bifurcation and chaos is expatiated and the numerical methods for analysis chaos are investigated. Theφ~4-Van der Pol-Duffing coupled system, a typical nonlinear oscillator equation is treated as a numerical example to state and verify different methods of the nonlinear time series analysis. A new analytic method is presented based on the averaging method. The region of a set of parameters are selected which can conduce chaos ofφ~4-Van der Pol-Duffing coupled system with cubic coefficient by the new analytic method. The parameters region is smaller and more accurate than that of Melnikov method. So the new analytic method has practical application and is worthy of studying further.3. The global bifurcation of soft nonlinear Duffing system and the symmetry-breaking bifurcation of hard nonlinear During systems are researched mainly. The chaotic attractors of nonlinear Duffing system are demonstrated by using an expanded software system and the oscillator circuit is designed by using EWB software for hardware simulation. Some new kinds of modes on the routes from symmetry-breaking bifurcation to periodic-doubling bifurcation are presented, which bridge the gap of two ways which discovered and studied before. One way is that hard Duffing system can lead to periodic-doubling bifurcation directly only via a symmetry-breaking bifurcation, and the other is that hard Duffing system can lead to periodic-doubling bifurcation via two symmetry-breaking bifurcations and an antisymmetry-breaking bifurcation. The relationship and distinguish between these two ways are clarified.4. The mechanism of time delayed feedback control of nonlinear Duffing systems is researched mainly. Unfortunately, the methods of ascertaining time feedback control parameters in autonomous systems are not applicable for non-autonomous chaotic systems anymore, but one can select the proper time feedback control parameters by Melnikov method, even which also has many shortcomings. And the correctness and the effectiveness of this method are illustrated by phase portraits and Lyapunov-exponent spectra. The oscillator circuit for hardware simulating the time delayed systems is designed by using EWB software.