Elementary Bifurcations of Non-Critical but Non-Hyperbolic Invariant Tori

Consider the time-periodic perturbations of n-dimensional autonomous systems with nonhyperbolic but non-critical closed orbits in the phase space. The elementary bifurcations, such as the saddle-node, transcritical, pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space (x, t), are studied. Some conditions which depend only on the original systems and can be used to determine the bifurcation structures of these problems are obtained. The theory is applied to two concrete examples.

Bifurcations of Invariant Tori and Subharmonic Solutions for Periodic Perturbed Systems

The time-periodic perturbations of planar Hamiltonian systems are investigated.A necessary condition for the existence of an invariant torus,a sufficient condition for the bifurcation of a unique invariant torus and a subharmonic solution are obtained.

受面内激励和横向外激励联合作用下蜂窝夹层板的双Hopf分叉

随着航空航天事业的发展,对各种材料性能的要求也越来越高.而蜂窝夹层板在结构和性能上具有许多优点,已在航空航天等领域应用广泛,并在一些重要结构中充当承力部件,但由于其特殊的蜂窝结构,相对于一般的板,在受力时会发生比较大的变形,所以用非线性理论研究蜂窝夹层板结构,并考察不同参数对非线性振动特性的影响,具有重要的理论和实际意义.如今,蜂窝夹层板的几何非线性问题已引起更多学者的关注.在一般均质理论的假设下,一些学者已经研究了各向同性蜂窝夹层板板的非线性动力学特性.本文研究了一类受面内激励和横向外激励联合作用下的四边简支蜂窝夹层板在主参数共振-1:2内共振时的双Hopf分叉问题.首先利用多尺度法得到系统的平均方程,然后结合分叉理论得到了系统的分叉响应方程,根据对分叉响应方程的分析,得到了六种不同的分叉响应曲线并给出了系统产生双Hopf分叉的条件.利用数值方法得到系统在参数平面的分叉集,通过对不同分叉区域的分析发现,随着参数的变化系统平衡点会分叉为两类周期解,随后周期解会通过广义静态分叉为准周期解,或者通过广义Hopf分叉为3D环面.

非双曲不变环面的初等分支

在相空间中 ,未扰动的n维自治系统 x =F(x)具有一个非双曲闭轨 .利用Floquet理论与平均法 ,讨论在周期扰动下此未扰动系统的非双曲不变环面在扩展相空间中的初等分支 .

一类非双曲不变环面的分支

未扰动的n维系统x=f(x),x∈Rn具有一个非双曲闭轨。利用Floquet理论及平均法,给出在周期扰动下,此非双曲闭轨的分支。

非双曲临界不变环面的分支

考虑一个在相空间具有非双曲临界闭轨Γ0的n维自治系统x.=F(x)的周期扰动系统x.=F(x)+G(t,x,α),应用比例变换和平均理论研究了该自治系统在空间(x,t)上的非双曲临界不变环面的分支,得到了系统x.=F(x)+G(t,x,α)存在不变环面的2个主要的结果.最后给出两个实例证明了这两个主要结果.

HOPF-BIFURCATIONS FOR NONHYPERBOLIC INVARIANT TORI

Consider the time-periodic perturbations of an n-dimensional autonomous system with a nonhyperbolic closed orbit in the phase space. By the method of averaging and Floquet theory, the bifurcations of the invariant torus in the extended phase space are studied.

Bifurcation theory of invariant tori of planar periodic perturbed systems

The existence conditions of an invariant torus for general planar periodic perturbed systems are given.It is proved that the perturbed systems may have not only large invariant tori which appear in pair,but also small invariant tori.

三维系统余维二分支中周期轨道与不变环面的存在性

本文研究具余维二奇点的三维系统在双参数扰动下周期轨道与不变环面的分支。