Double Hopf Bifurcation Theory In Nonlinear Dynamical Systems And Applications In Engineering Systems

Hopf bifurcation is a relatively simple but very important dynamic bifurcaton. It isnot only used to study the dynamic bifurcation and limit cycle, but also is also closelylinked with the generation mechanism of self-excited vibration. Hopf bifurcation theoryhas become a classic tool to study differential equations small amplitude periodic solutionand demise. Research of Hopf bifurcation has important theoretical significance. Researchof the double Hopf bifurcation is a very important issue, which being on the practicalproblems in the high-dimensional nonlinear dynamic equations in the engineering field.The results of our researching mainly include the promotion of high.dimensionalHopf bifurcation theorem, so that it can be used to study the nonlinear dynamical systemsin the resonance case. Using the Hopf bifurcation theory, the double Hopf bifurcation of aclass of honeycomb sandwich plates was studied, which is under the combined effects ofplane and transverse incentive. Subsequently, appling the multi.scale methods and theHopf bifurcation theory, a class of composite laminated plates double Hopf bifurcationwas studied, and gave the parameter diagrams on the parameter plane. Furthermore,studied the singularity theory, and used it to giving the explaining of the double Hopfbifurcation of piezoelectric laminated composite plate under the bombined effects of theplane and transverse and gave the different forms of movement in different regions.The main research contents obtained in this dissertation are as follows:(1) We gave a further study of the high-dimensional Hopf bifurcation theorem, andstudied the double Hopf bifurcation of the honeycomb sandwich plate system as theprimary parametric-1:2internal resnonace. First, by using the method of multiple scales,we studied the averaged equations of the plate system in the form of Cartesian and polarcoordinates. Based on the averaged equations, we gave the parameter bifurcation biagram.The numerical simulation gave the local bifurcation diagram of the system, and sometypes of periodic motions of the honeycomb sandwich plate system.(2) Using the Hopf bifurcation theorem aboute the high-dimensional nonlinearsystem, we studied the double Hopf bifurcation of the laminated composite plate subjectto the surface incentives and lateral excitation, which is in the primaryparametric-1:1internal resonace. Consider the case of weak damping and weak parametricexcitation, we gave the two coordinate forms of the average equation of the system, andstudied the bifurcation response. The parameter bifurcation diagram of the system wasobtained.(3) Extended the singularity theory of nonlinear systems as in non.resonant case tothe resoant case. By introducing the linear transformation, we proved the bud equivalencetheorem. Then we gave the concepts of unfolding and universal unfolding. We studied the double Hopf bifurcation of the nonlinear system in different resonance cases. The systemmay present periodic motion or quasi-periodic motion.(4) Based on the theoretical results of the fourth chapter, we analysis the double Hopfbifurcation of the piezoelectric composite laminated plate as in the primary parameter-1:3internal resonant. Using the method of multiple scales we derived the averaged equationsof the system. Appling the results in4.2, we studied the different bifurcation zones in theparameter bifurcation diagram. We found the euqlibrium solution of the system is instable,and may bifurcate to different types of periodic solutions with different conditions.In closing remarks, full text summarizes and the further research directions wereproposed.