参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉

在本文里我们首先研究了具有Z_2-对称性的范式理论和退化向量场的普适开折理论。然后利用上述理论研究了参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉,从而解决了当解具有两个零特征值时解的稳定性问题。最后利用Melnikov方法求出了参数平面上的同宿分叉曲线,讨论了全局分叉的存在性。     

非线性参数激励系统的动力分叉研究

本文针对弹性梁动力曲屈分叉问题,建立了系统的非线性Mathiue方程,较全面地讨论了此类参数激励系统的1/2亚谐分叉特性,指出以往对此类问题的研究得到的只是一种退化情形下的分叉特性,阐述了分叉方程的截断对分叉结果的影响,得到了一些新的结果。文中还介绍了一个模型弹性梁系统分叉响应特性的实测结果,证实了理论分析的可靠性。     

非线性周期参激系统的1：3共振分叉

本文从群的观点出发，建立了Ｚ３－等变的奇点理论。利用这个结果，我们讨论了非线性参数激励系统－－Ｍａｔｈｉｅｕ方程的１：３共振分叉。给出了非退化民政部下的全体分 图。数值模拟验证了我们的理论结果。     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。     

一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉

通过理论分析和数值仿真,研究了一类二维冲击振动系统在一种强共振条件下的Hopf分叉与亚谐分叉。分析并证实了该类系统在此共振条件下可由稳定的周期1 1振动分叉为周期4 4振动或概周期振动,讨论了亚谐振动和概周期振动向混沌运动的演化过程。     

悬索的多重内共振研究

本文对谐波激励的悬索的非线性响应进行了研究,同时考虑了如下问题(1):面内第三阶对称模态的主共振:(2):面内第一阶、第三阶对称模态和面外第五阶模态之间的内共振.本方首先针对考虑大变形的悬索动力学方程,由线性理论求得各阶频率,考察可能出现的内共振.然后利用直接法对悬索的运动学方程和边界条件进行非线性求解.由多尺度法得到系统的平均方程和悬索响应的二阶近似解.随后利用Newton-Raphson 方法和弧长法对特定张拉索进行数值仿真计算,得到面内第一阶对称模态、面内第三阶对称模态和面外第五阶模态的稳态解,并分析了解的稳定性.绘制幅频响应曲线,发现了关于悬索响应的多种分叉现象,并且对各种分叉现象周期解、混沌解进行了讨论.     

对称铺设正交各向异性层合板的亚谐参数共振

本文应用奇异性理论讨论了对称铺设正交各向异性层合矩形板的亚谐参数共振问题。主要内容是用Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ方法结合Ｚ２－对称等变的概念，使分叉方程转化为代数方程的研究，同时给出了参数平面上不同参数域中各种可能的分叉曲线。     

具有单侧刚性约束的两自由度振动系统在强共振条件下的拟周 …

采用理论分析和数值仿真相结合的方法,研究了一类两自由度碰撞振动系统在一种强共振条件下的Ｈｏｐｆ分叉问题,分析并证实了碰撞振动系统在此共振条件下可由稳定的周期１－１振动分叉为不稳定的周期３－３振动,讨论了亚谐振动向混沌运动的演化过程。     

具有单侧刚性约束的两自由度振动系统在强共振条件下的拟周期运动与混沌

采用理论分析和数值仿真相结合的方法,研究了一类两自由度碰撞振动系统在一种强共振条件下的Hopf分叉问题.分析并证实了碰撞振动系统在此共振条件下可由稳定的周期1-1振动分叉为不稳定的周期3-3振动,讨论了亚谐振动向混沌运动的演化过程.     

一类强非线性振动系统的分叉

对于参数激励和强迫激励共同作用的一类强非线性系统，本文先用改进的Ｌ－Ｐ方法求出了变换参数，使该系统的解能展为小参数的幂级数．然后利用多尺度法求出了该系统的分叉响应方程．研究了这类强非线性系统的余维１分叉问题，画出了转迁集和分叉图     

A torus bifurcation theorem with symmetry

A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D₃) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.     

Existence of 3-Dimensional Tori for 1D Complex Ginzburg–Landau Equation Via a Degenerate KAM Theorem

In this paper, we consider complex Ginzburg–Landau equation in one space dimension and rigorously show the existence of 3-dimensional tori. The proof is based on degenerate infinite-dimensional KAM theory and normal form technique.     

Existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.     

Navier–Stokes Equations: Controllability by Means of Low Modes Forcing

We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and continuous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.     

Stability of Solutions of a Degenerate Linear System of Differential Equations

We introduce the concept of stability of solutions of a system of linear differential equations with an identically degenerate matrix as the coefficient of the derivative. We find necessary and sufficient conditions for the stability of such systems. We generalize the Floquet–Lyapunov theory to systems of this type with periodic coefficients.     

Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations

A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger’s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.     

Adjoint operator method and normal forms of higher order for nonlinear dynamical system

I.IntroductionInordertostudybifurcationsofnonlineardynamicalsystemsinthedegeneratecasesofhighercodimensionnumber(>3),wemustcomputenormalformsofhigherorderfornonlineardynamicalsystems.Inrecenttwentyyearsmanyscientistsmadeveryimportantcontributionstodevelop…     

悬臂输流管道的运动分岔现象和混沌运动

研究受约束悬臂输流管道的稳定性和运动分岔问题.在静态与动态失稳区域边界上的一个交叉点附近,用理论分析的方法详细研究了该系统可能发生的复杂运动和运动分岔现象.在动态失稳区域内发现了输流管道的概周期运动和由于概周期运动环面破裂而导致混沌的现象.理论分析结果与数值模拟结果相吻合.     

Linear Potentials in Nonlinear Potential Theory

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C ¹-level.