Hopf And Saddle-Node Bifurcation Control In Multiple-degree-of-freedom Nonlinear Systems

Control of bifurcation and chaos in nonlinear differential dynamic systems has recently caused great interests in researchers. Compared to chaos control, reports on the former were much less. Bifurcation is a specific phenomenon which generally existed in such fields as mechanics, physics, chemistry, medical, biology and economics. Bifurcation control plays a vital roll in bifurcation research, which means a controller should be recalled to change the characteristic of bifurcation. Typical bifurcation control includes: delay appearance point of bifurcation, generate a new bifurcation, change parameter of a bifurcation point, stabilize bifurcation solution, control the multiplicity/ amplitude/frequency of a limit cycle, optimize system behavior around bifurcation point, reduce instable zone and so on. In engineering, bifurcation control is to avoid harmful kinetics behavior and take supervisory to the system. Hopf bifurcation is an important dynamic bifurcation in bifurcation research and saddle-node bifurcation is one of three rudimentary static bifurcations. Reports on multiple-degree-of-freedom nonlinear systems were still minor at the present time.The paper did a systematic and profound research in control of bifurcation and chaos based on mathematical theory, thus, set a theoretical foundation for its application in engineering projects. Some innovative conclusions are drawn as follows:1. Applied feedback control method in coupled multiple-degree-of-freedom nonlinear van der Pol systems to control Hopf bifurcation and its amplitude, control strategy and method were achieved.2. Applied feedback control method to control the saddle-node bifurcation in complicated multiple-degree-of-freedom nonlinear systems, revealed the relationship of controlling parameter and the instable intervals.3. Proceed anti-chaos control in coupled multiple-degree-of-freedom nonlinear van der Pol systems, produced a symmetric chaotic attractor, thus, made anti-chaos control in high dimensional nonlinear coupled van der Pol system possible.4. Studied a three-dimensional system derived from the Chen system, obtained its chaotic attractor, applied feedback controller to perform chaos control of the system, and finally contract the system moving to a equilibrium point.