广义同步化流形的Holder连续性

证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是Holder连续的.采用的思想是Temam的无穷维动力系统的惯性流形理论的改进.在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义在一个函数类上的映射满足Schauder不动点定理,从而得到广义同步化流形,该广义同步化流形具有不变性.又证明了存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性.数值仿真证实了理论的正确性.在驱动系统和响应系统外引入辅助系统,辅助系统与响应系统的完全同步化表明了驱动系统和响应系统的广义同步化.     

广义同步化流形的Hlder连续性

证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是Hlder连续的.采用的思想是Temam的无穷维动力系统的惯性流形理论的改进.在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义在一个函数类上的映射满足Schauder不动点定理,从而得到广义同步化流形,该广义同步化流形具有不变性.又证明了存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性.数值仿真证实了理论的正确性.在驱动系统和响应系统外引入辅助系统,辅助系统与响应系统的完全同步化表明了驱动系统和响应系统的广义同步化.     

非自治Schrdinger-KdV型藕合方程组的一致吸引子及其维数估计

本文研究了非自治Schr■dinger-KdV型藕合方程组的非线性动力学行为．运用具有两个参数的算子簇来描述非自治无穷维动力系统的方法,证明了该系统的一致吸引子的存在性．进一步,对其Hausdorff维数进行了估计．     

非自治Schrdinger-KdV型藕合方程组的一致吸引子及其维数估计

本文研究了非自治Schrodinger-KdV型藕合方程组的非线性动力学行为．运用具有两个参数的算子簇来描述非自治无穷维动力系统的方法，证明了该系统的一致吸引子的存在性．进一步，对其Hausdorff维数进行了估计．     

非线性发展方程的H(o)lder连续惯性流形

引入非线性发展方程的H\"older连续惯性流形的概念,为原来惯性流形概念的推广和修正.惯性流形是有限维不变的Lipschiz流形,是研究发展方程解的长时间性态的合适工具,其缺点是需要谱间隙条件.提出H\"older连续惯性流形也是有限维不变的,但光滑性减弱为H\"older连续,不需要谱间隙条件.该流形与指数吸引子交集具有指数吸引性,无穷维动力系统可在H\"older连续惯性流形上约化为有限维常微分方程组.     

Banach空间非线性脉冲Volterra积分方程组的整体解

研究Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的非线性脉冲Volterra积分方程组解的存在性。给出了若干极值解的存在定理,改进了定义在有限区间上具有有限个脉冲点情形时该类方程的相应结果,并利用该结果讨论了一个无穷维积分方程组。     

弱阻尼KdV方程中长期动力学行为研究*

证明了周期边界条件下弱阻尼KdV方程存在近似惯性流形．该流形使吸引子被确切定义的有限维光滑流形逼近．并由此概念引出新的数值方法很好地用于研究动力系统长期行为．     

离散扰动NLS方程组的Smale马蹄与混沌(Ⅰ)——Poincaré映射

利用n维Conley-Moser条件证明了一类离散扰动非线性Schrdinger方程(NLS)的Smale马蹄的存在性．由以上结果,我们得到离散扰动NLS方程组存在不变集Λ,其动力系统与四符号变换拓扑共轭．     

离散扰动NLS方程组的Smale马蹄与混沌(Ⅱ)——Smale马蹄

利用n维Conley-Moser条件证明了一类离散扰动非线性Schrdinger方程(NLS)的Smale马蹄的存在性．由以上结果,我们得到离散扰动NLS方程组存在不变集Λ,其动力系统与四符号变换拓扑共轭．     

半离散Schr?dinger-BBM型方程的全局吸引子

主要研究用Crank-Nicolson格式对时间t半离散化的Schr?dinger-BBM方程组的长时间行为,证明了该半离散化方程全局吸引子的正则性.首先证明半离散方程在H~1×H~1空间上生成一个离散无穷维动力系统,并且在H(3/2-ε)×H~2拥有一个全局吸引子A_τ;然后证明该全局吸引子A_τ是正则的,即A_τH~(3/2-ε)×H~2是有界的并且是紧的.     

Kolmogorov流动模型的七模截断及其混沌特性

为了给出Kolmogorov流动模型中混沌行为的数学描述,选取常数k=3,重新对描述该模型的Navier-Stokes方程进行截断,得到了一个新的七维混沌系统．数值模拟了控制参数在一定范围内变化时方程组的基本动力学行为和混沌轨线,分析了其混沌特性．一方面证实了具有湍流特性的数学对象归因于低维混沌吸引子,另一方面有利于更好地了解湍流流动产生的机理.     

Short chaotic strings and their behaviour in the scaling region

Coupled map lattices are a paradigm of higher-dimensional dynamical systems exhibiting spatio-temporal chaos. A special case of non-hyperbolic maps are one-dimensional map lattices of coupled Chebyshev maps with periodic boundary conditions, called chaotic strings. In this short note we show that the fine structure of the self energy of this chaotic string in the scaling region (i.e. for very small coupling) is retained if we reduce the length of the string to three lattice points.     

Global finite-time synchronization of different dimensional chaotic systems

This paper addresses the problem of global finite-time synchronization of two different dimensional chaotic systems. Firstly, the definition of global finite-time synchronization of different dimensional chaotic systems are introduced. Based on the finite-time stability methods, the controller is designed such that the chaotic systems are globally synchronized in a finite time. Then, some uncertain parameters are adopted in the chaotic systems, new control law and dynamical parameter estimation are proposed to guarantee that the global finite-time synchronization can be obtained. By considering a dynamical parameter designed in the controller, the adaptive updated controller is also designed to achieve the desired results. At last, the results of two different dimensional chaotic systems are also extended to two different dimensional networked chaotic systems. Finally, three numerical examples are given to verify the validity of the proposed methods.     

On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems

For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.     

On stochastic and chaotic motion

In this paper we prove results regarding certain precise relationships between random motion and chaotic motion. In particular we prove a strong invariance principle for smooth functions of certain chaotic dynamical systems, and show that solutions of dynamical systems which are coupled to such chaotic systems may be approximated by solutions of stochastic differential equations     

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

Chaos,solitons and fractals in hidden symmetry models

A spontaneous symmetry breaking (or hidden symmetry) model is reduced to a system nonlinear evolution equations integrable via an appropriate change of variables, by means of the asymptotic perturbation (AP) method, based on spatio-temporal rescaling and Fourier expansion. It is demonstrated the existence of coherent solutions as well as chaotic and fractal patterns, due to the possibility of selecting appropriately some arbitrary functions. Dromion, lump, breather, instanton and ring soliton solutions are derived and the interaction between these coherent solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one can construct lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.     

离散的Burgers-Ginzburg-Landau方程组的吸引子

文章通过对空间变量的有限差分方法离散了具有周期边值的Ｂｕｒｇｅｒｓ Ｇｉｎｚｂｕｒｇ Ｌａｎｄａｕ方程组．研究了这个离散方程组初值问题解的适定性．证明了当差分网格足够大时离散方程组存在吸引子，并得到了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计．这个上界不会随着网格的加细而无限增大，因此数值分析离散的有限维系统的吸引子可以近似探讨原无限维系统的吸引子．     

On Multi‐component Ermakov Systems in a Two‐Layer Fluid: Integrable Hamiltonian Structures and Exact Vortex Solutions

By introducing an elliptic vortex ansatz, the 2+1‐dimensional two‐layer fluid system is reduced to a finite‐dimensional nonlinear dynamical system. Time‐modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen‐Ermakov‐Pinney equation.