Limit cycles in the equation of whirling pendulum with autonomous perturbation

The two-parameter Hamiltonian system with the autonomous perturbation is considered. Via the Mel'nikov method, existence and uniqueness of a limit cycle of the system in a certain region of a two-dimensional space of parameters is proved.     

Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions

This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincaré section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.     

A coaxial vortex ring model for vortex breakdown

A simple-yet plausible-model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about all of the characteristics that have been observed in what is now a substantial body of published research on the phenomenon of vortex breakdown. Moreover, it is demonstrated how the very nature of the fluid dynamics in axisymmetric breakdown flows can be predicted and controlled by the choice of the initial ring configurations and their vortex strengths. The dynamic intricacies produced by the two ring + swirl model are illustrated with several numerical experiments.     

On the Dynamics of Navier-Stokes and Euler Equations

This is a detailed study on certain dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a) zero viscosity limit of the spectra of linear Navier-Stokes operator, (b) heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Due to the difficulty of the problem for the full Navier-Stokes and Euler equations, we also propose and study two simpler models of them.     

带慢变角参数摄动平面非Hamilton可积系统的混沌

将Melinikov方法推广到带慢变角参数摄动平面可积系统。基于对未受摄动系统几何结构的分析,建立了横截同宿条件。借助常微分方程组解对参数的可微性定理,得到系统的广义Melnikov函数,其简单零点意味着系统可能出现混沌。     

带偏心转子陀螺体的混沌运动

本文讨论了无力矩条件下带有质量偏心轴对称转子的非对称陀螺体的运动。利用动量变量列写动力学方程,并将系统化作受周期微扰作用下的Ｅｕｌｅｒ－Ｐｏｉｎｓｏｔ运动。应用Ｍｅｌｎｉｋｏｖ方法预测系统存在Ｓｍａｌｄ马蹄意义下的混沌运动,此结论与Ｐｏｉｎｃａｒｅ截面的数值计算相符。从Ｐｏｉｎｃａｒｅ截面的相图也可明显看出转子对于双自旋卫星的姿态稳定作用。     

MELNIKOV FUNCTIONS AND PERTURBATION OF A PLANAR HAMILTONIAN SYSTEM

In this paper, Melnikov functions which apper in the study of limit cycles of a perturbed planar Hamiltonlan system are studied. There are two main contributions here. The first one is related to the explicit formulae for these functions: a new method is developed to achieve the goal for the second one (Theorem A). the authors also discover a close relation between Melnlkov functions and focal qtmntities (Theorem 13). This relation is useful in both judging when a Melnikov function is identically zero and simplifying the computation of a Melnikov function (see §5). I)espite these results, this paper also includes other related resuEs, e.g. some estimations of the upper bound for the number of limit cycles in a perturbed Hamiltonian system.     

Quasi-Periodic Oscillations,Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.     

Sequences of orbits and the boundaries of the basin of attraction for two double heteroclinic orbits

The boundaries of the basin of attraction are usually assumed to be rather elementary for Hamiltonian systems with autonomous perturbations. In the case of one saddle point, the sequences of orbits before capture are unique for each basin. However, we show that for two saddle points each with double heteroclinic orbits, there is an infinite number of different sequences of nearly homoclinic orbits before capture depending on the four heteroclinic parameters. The probabilities of capture are independent of the capture sequence.     

用集结坐标法对细杆中呼吸子状态的分析

研究了计入Peierls-Nabarro(P-N)力和材料粘性效应的一维无限长金属杆在简谐外力扰动下的动力响应,导出了类sine-Gordon 型的运动方程．在集结坐标(collective coordinate）下原控制方程可以用常微分动力系统描述,研究系统中呼吸子的运动．根据非线性动力学方法分析,P-N力的幅值和频率的变化将改变双曲鞍点的位置,并改变系统次谐分叉的阈值,但不改变由奇阶次谐分叉通向混沌的路径．通过实例给出了P-N力幅值和P-N力频率对细杆动力响应的详细影响过程,可见混沌发生的区域是一个半无限区域,并随着P-N力的增大而增大．P-N力的频率对系统有类似的影响．