Melnikov函数和Poincaré映射

本文中我们给出了Melnikov函数和Poincaré映射的关系,从而给出了Melnikov方法的新的证明.本文的优点是给出了更明确的解,并把次谐分支的Melnikov函数与稳定流形与不稳定流形横截相交的Melnikov函数统一成为一个公式.     

Bouncing Ball模型的弱混沌性

用异于传统的方法，作出Bouncing Ball映射不变流形的对称流形，从而成功地将稳定流形与不稳定流形的位置进行比较。应用[1]关于弱横截与弱混沌的有关概念及定理，给出了Borncing Ball映射产生弱混沌的较为一般的参数区域，进一步提示了Bouncing Ball映射的动力学行为。     

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下,利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式,利用广义Melnikov方法研究该系统的多脉冲混沌动力学,并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下,提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下,矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．     

一类带有慢变参数的sine-Gordon方程的单脉冲异宿轨道

基于Fenichel的几何奇异摄动理论,结合Melnikov方法,该文研究一类带慢变参数的sine-Gordon方程单脉冲波前解的存在性.首先,基于几何奇异摄动理论进行快慢分离,获得层系统和退化系统及其动力学;接着,引入Melnikov函数度量慢流形的稳定和不稳定流形的横截相交性,获得Take-off和Touch-down曲线的解析式.控制Take-off和Touch-down曲线使之分别与两个慢流形上鞍点的不稳定和稳定流形横截相交,从而得到奇异异宿轨道的存在性.经摄动,在该奇异异宿轨附近可获得异宿于系统两个不同鞍点的异宿轨道的存在性,从而上述带慢变参数的sine-Gordon方程的单脉冲波前解的存在性可得.最后,考虑了一个具体的例子,验证理论结果的正确性.     

具有暂时免疫传染病模型同宿分支

讨论了具有暂时免疫传染病模型同宿轨道分支的存在性,利用Melnikov函数确定了系统双曲不动点的稳定和不稳定流形的相对位置,从而给出存在极限环的参数范围.     

软弹簧型Duffing方程在摄动下分支出的极限环

在这篇文章中,作者用Melnikov函数方法分析了软弹簧型Duffing方程[1]在摄动下异宿轨道破裂后稳定流形与不稳定流形的相对位置,给出了方程在不同摄动下分支出极限环的条件与极限环的位置.     

托卡马克中低模态到高模态转迁模型的混沌运动

利用Melnikov方法详细研究了在托卡马克(Tokamaks)中,等离子区边缘附近低模态到高模态转迁方程的混沌动力学.该转迁方程是一个含外激励和参数激励的系统.对含周期外激励和线性参数激励、三次参数激励的系统分别绘出了用来划分混沌区和非混沌区的临界曲线.得到的结果表明,含有线性或三次参数激励的系统存在不可控区域,在该区域中异宿轨分岔总是导致混沌发生.特别地,三次参数激励系统存在一个"可控频率",施以该频率的激励,不论激励的振幅多大,同宿轨分岔总是不会导致混沌发生.得到了这类系统的一些复杂的动力学行为.     

指数二分性与奇异摄动系统的横戳异宿轨道

本文研究奇异摄动系统的横截异宿轨道的存在性,利用指数二分性理论和Liapunov－Schmidt方法,获得了判断奇异摄动系统存在横截异宿轨道的Melnikov型函数,因而推广了一些文献的结果．     

超临界情形的Melnikov函数及应用

该文具体推导了三阶Melnikov函数的积分表达，解决了电机工程中提出的一类系统（见[5]），当参数时的超临界（一阶、二阶Melnikov函数恒为零）的情形下，系统的稳定流形与不稳定流形的相对位置的确定问题．并通过环面上的VanderPol方程，对[2]与[4]所给的二阶Melnikov函数的表达式进行了比较，结果发现[2]所给的平面自治系统的二阶，n阶表达式均是错的.该文在最后作了纠正.     

同宿流形中的奇异轨道

研究较一般的高维退化系统的同宿、异宿轨道分支问题．利用推广的Melnikov函数、横截性理论及奇摄动理论，对具有鞍—中心型奇点的带有角变量的奇摄动系统，在角变量频率产生共振的情况下，讨论其同宿、异缩轨道的扰动下保存和横截的条件．推广和改进了一些文献的结果。     

Homoclinic Tangles, Phase Locking, and Chaos in a Two-Frequency Perturbation of Duffing's Equation

Summary. We study a two-frequency perturbation of Duffing's equation. When the perturbation is small, this system has a normally hyperbolic invariant torus which may be subjected to phase locking. Applying a version of Melnikov's method for multifrequency systems, we detect the occurrence of transverse intersection between the stable and unstable manifolds of the invariant torus. We show that if the invariant torus is not subjected to phase locking, then such a transverse intersection yields chaotic dynamics. When the invariant torus is subjected to phase locking, the situation is different. In this case, there exist two periodic orbits which are created in a saddle-node bifurcation. Using another version of Melnikov's method for slowly varying oscillators, we also give conditions under which the stable and unstable manifolds of the periodic orbits intersect transversely and hence chaotic dynamics may occur. Our results reveal that when the invariant torus is subjected to phase locking, chaotic dynamics resulting from transverse intersection between its stable and unstable manifolds may be interrupted. Received November 18, 1993; final revision received September 9, 1997; accepted October 27,1997     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed.     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

     

Horseshoes for the nearly symmetric heavy top

We prove the existence of horseshoes in the nearly symmetric heavy top. This problem was previously addressed but treated inappropriately due to a singularity of the equations of motion. We introduce an (artificial) inclined plane to remove this singularity and use a Melnikov-type approach to show that there exist transverse homoclinic orbits to periodic orbits on four-dimensional level sets. The price we pay for removing the singularity is that the Hamiltonian system becomes a three-degree-of-freedom system with an additional first integral, unlike the two-degree-of-freedom formulation in the classical treatment. We therefore have to analyze three-dimensional stable and unstable manifolds of periodic orbits in a six-dimensional phase space. A new Melnikov-type technique is developed for this situation. Numerical evidence for the existence of transverse homoclinic orbits on a four-dimensional level set is also given.     

A Geometric Construction of Traveling Waves in a Bioremediation Model

Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding to the motion of a biologically active zone, in which the microorganisms consume both substrate and acceptor. For certain values of the parameters, the traveling waves exist on a three-dimensional slow manifold within the five-dimensional phase space. We prove persistence of the slow manifold under perturbation by controlling the nonlinearity via a change of coordinates, and we construct the wave in the transverse intersection of appropriate stable and unstable manifolds in this slow manifold. We study how the TW depends on the half-saturation constants and other parameters and investigate numerically a bifurcation in which the TW loses stability to a periodic wave.     

Self-oscillations of a third order PLL in periodic and chaotic mode and its tracking in a slave PLL

The dynamics of a third order phase locked loop (PLL) with a resonant low pass filter (LPF) has been studied numerically in the parameter space of the system. The range of stable synchronous operating zone of the PLL, expressed in terms of system and signal parameters, is estimated. The obtained results are in agreement with the analytically predicted results in the literature. The PLL dynamics in the unstable region is found to have a sequence of period doubling bifurcation and chaos. In the master–slave mode of operation of two 3rd order PLLs, the slave PLL can track the periodic as well as chaotic dynamics of the master PLL for a narrow range of effective frequency offset when other design parameters are within the stable zone as predicted for an isolated PLL. The synchronization of the master and slave PLLs in this condition is proved to be a generalized one using the auxiliary slave system approach. Experimental observations on prototype hardware circuits for an isolated PLL and for a master–slave PLL arrangement are also given.