The Research Of Complex Dynamics Of Some Nonlinear Systems

The structure models of rectangular plates and buckled beams are widely used in engineering fields.Most of these models can be described by the high-dimensional even infinite dimensional nonlinear dynamical equations.Numerical simulation can reveal the complicated nonlinear dynamical behavior of these models.But it is more difficult to investigate the high-dimensional nonlinear dynamical system in theoretical method and space geometrical description.Theoretical research of nonlinear dynamical systems have developed from low dimensional to high dimensional.Bidurcations and chaotic dynamics of high dimensional nonlinear dynamics have become the major subject in nonlinear science.Some analytical methods including normal form method,global perturbation method,energy-phase method and extended Melnikov method are applied to several types of high dimensional nonlinear models to reveal the complex dynamic behavior.The main contents of this paper are as follows.(1)The stability and bifurcation behaviors for a cantilever functionally graded materials rectangular plate subjected to the transversal excitation in thermal environment are studied by means of combination of analytical and numerical methods.Four types of degenerated equilibrium points are studied in detail.For each case,choosing (?1,?2)as perturbation parameter,the stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained,which may lead to static bifurcation and Hopf bifurcation.The phase portraits of the degenerated equilibrium points are obtained by numerical simulation which shows the motion state of this kind of cantilever plate model under different parameters.(2)The global bifurcation and chaotic dynamics of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations are investigated by using the global perturbation method and the energy-phase method.The explicit sufficient conditions for the existence of the Shilnikov-type single-pulse and multi-pulse homoclinic orbits are obtained,which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes.The numerical results demonstrate that there exists chaos of the composite laminated cantilever rectangular plate,in addition damping and excitation parameters have important influence on chaotic motion of the system.(3)The global bifurcation and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in-plane excitations are investigated with the case of principal parametric resonance.The existence of Shilnikov-type single-pulse homoclinic orbits and chaotic dynamics are investigated by using the global perturbation method.The extended Melnikov method is used to investigate the Shilnikov-type multi-pulse homoclinic orbits and chaotic dynamics in the resonance case.The multi-pulse Melnkov function and several sufficient conditions of chaos are given.(4)The existence of homoclinic orbits and heteroclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation with 1:1 internal resonance is studied.Each bump is a fast excursion away from the resonance band,and the bumps are interspersed with slow segments near the resonance band.The homoclinic orbits and homoclinic orbits are composed of alternating slow and fast pieces which include multi-bump Shilnikov orbits,connect equilibria and periodic orbits in the resonance band.(5)The subharmonic bifurcations and chaos for one kind of buckled beam model subjected to parametric excitations are investigated.The critical curves separating the chaotic and non-chaotic regions are obtained by utilizing Melnikov method.The conditions for subharmonic bifurcations are also obtained.