Chaotic behavior in the nonlinear elastic beam

The dynamic response of the non-linear elastic simply supported beam subjected to axial forces and transverse periodic load is studied. Melnikov method is used to consider the dynamic behavior of the system whose post-buckling path is steady. The effect of the higher order terms in the controlling equation is taken into account. It is found that the fifth-order terms have a great influence on the dynamic behavior of the system. The result shows that there exist either homoclinic orbits or heteroclinic orbits in the system. In this paper, the critical values of the system entering chaotic states are given. The diagram of an example is shown. The project is supported by the National Natural Sciences Foundation of China.     

Exponential dichotomies and transversal homoclinic orbits in degenerate cases

In this paper, we first give a sufficient condition which assures that a linear differential equation depending on a small parameter admits an exponential dichotomy onR, then we use the result obtained here on exponential dichotomies to investigate the existence of transversal homoclinic orbits of perturbed differential systems in two degenerate cases and obtain a Melnikov-type vector. The results on exponential dichotomies of this paper provide us a tool of proving the transversality of homoclinic orbits in studying degenerate bifurcations.This work is supported by NSF of China.     

一类慢变参数振子系统的同宿分叉及其安全盆侵蚀

分析一个具有慢变参数的非线性系统,利用Ｍｅｌｎｋｏｖ方法,分析了系统在参数发生变化时的同宿分叉,同时利用分叉结果,数值讨论了当系统参数发生变化时安全盆的侵蚀及分叉,混沌的联系。     

Higher-order Melnikov method

In this paper the Melnikov method has been generalized to the case of higher-order byfinding an explicit expression for second-order subharmonic Melnikov function,and it hasbeen proved that the existence of subharmonic or hyper-subharmonic of a system can beproved under certain conditions by use of second-order Melnikov function.     

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。     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

A group of chaotic motion of soft spring quadratic duffing equations

In this paper,we use the Melnikov function method to study a kind of soft Duffing equations(?) Af((?),x) x-x~(2k 1)=r[M(x,(?))cosωt N(x,(?))sinωt](k=1,2,3…)and give the condition that the equations have chaotic motion and bifurcation.The method used in this paper is effective for dealing with the Melnikov function integral of the system whose explict expression of the homoclinic or heteroclinic orbit cannot be given.     

Chaos in Perturbed Planar Non-Hamiltonian Integrable Systems with Slowly-Varying Angle Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＭｅｌｎｉｋｏｖｍｅｔｈｏｄｆｏｒｄｅｔｅｃｔｉｎｇｃｈａｏｓ[1]ｈａｓｂｅｅｎｅｘｔｅｎｄｅｄｔｏｈｉｇｈ＿ｄｉｍｅｎｓｉｏｎａｌｓｙｓｔｅｍｓｗｉｔｈｓｌｏｗｌｙ＿ｖａｒｙｉｎｇａｎｇｌｅｐａｒａｍｅｔｅｒｓ ,ｂｕｔｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｕｎｐｅｒｔｕｒｂｅｄｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍｓａｒｅｒｅｑｕｉｒｅｄｔｏｂｅＨａｍｉｌｔｏｎｉａｎ[2 ].Ｆｏｒｐｅｒｔｕｒｂｅｄｐｌａｎａｒｎｏｎ＿Ｈａｍｉｌｔｏｎｉａｎｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍｓ,ｔｈ…     

BIFURCATION IN A PARAMETRICALLY EXCITED TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEM WITH 1∶2 INTERNAL RESONANCE

ＩｎｔｒｏｄｕｃｔｉｏｎＰａｒａｍｅｔｒｉｃａｌｙｅｘｃｉｔｅｄｖｉｂｒａｔｉｏｎｓｈａｖｅｂｅｎｅｘｔｅｎｓｉｖｅｌｙａｐｐｌｉｅｄｔｏｉｎｅｎｇｉｎｅｒｉｎｇｓｙｓｔｅｍｓ［１］,ｓｕｃｈａｓｔｈｅｖｉｂｒａｔｉｏｎｏｆｍａｃｈｉｎｅｓ,ｄｙｎａ...     

Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1∶2 internal resonance

The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos. Project supported by the National Natural Science Foundation of China (19472046)     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.     

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.     

Chaotic response of a large deflection beam and effect of the second order mode

This study intends to investigate the dynamic behavior of a nonlinear elastic beam of large deflection. Using the Galerkin principle, the dynamic nonlinear governing equations are derived based on the single and double mode methods. Two different kinds of nonlinear dynamic equations are obtained with the variation of the dimension and loading parameters. The chaotic critical conditions are given by Melnikov function method for the single mode model. The chaotic motion is investigated and the comparison between single and double mode models is carried out. The results show that the single mode method usually used may lead to incorrect conclusions in some conditions, and instead the double mode or higher order mode method should be used. Finally, the applicable condition of the single mode method is analyzed.     

A Simplified Optimal Control Method for Homoclinic Bifurcations

A simplified optimal control method is presented for controlling or suppressing homoclinic bifurcations of general nonlinear oscillators with one degree-of-freedom. The simplification is based on the addition of an adjustable parameter and a superharmonic excitation in the force term. By solving an optimization problem for the optimal amplitude coefficients of the harmonic and superharmonic excitations to be used as the controlled parameters, the force term as the controller can be designed. By doing so, the control gain and small optimal amplitude coefficients can be obtained at lowest cost. As the adjustable parameter decreases, a gain of some amplitude coefficient ratio is increased to the highest degree, which means that the region where homoclinic intersection does not occur will be enlarged as much as possible, leading to the best possible control performance. Finally, it is shown that the theoretical analysis is in agreement with the numerical simulations on several concerned issues including the identification of the stable and unstable manifolds and the basins of attraction.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

Chaotic pitch motion of an asymmetric non-rigid spacecraft with viscous drag in circular orbit

We study the pitch motion dynamics of an asymmetric spacecraft in circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by a small aerodynamic drag torque proportional to the angular velocity of the body about its mass center. We also suppose that one of the moments of inertia of the spacecraft is a periodic function of time. Under both perturbations, we show that the system exhibits a transient chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for heteroclinic chaos in terms of the system parameters which is numerically contrasted. We also show that some periodic orbits survive for perturbation small enough.     

白噪声参激余维2分叉系统研究

为考查白噪声参激的具有非半简双零特征值的一类余维二分叉系统的样本稳定性并确定其首次分叉点的位置，本文使用Ｌ．Ａｒｎｏｌｄ的渐近分析方法研究了系统最大Ｌｙａｐｕｎｏｖ指数的渐近分析式．为进一步研究噪声对于此系统退化分叉的影响，本文将使用一维扩散过程的奇异边界理论，考查“隐藏在余维２分叉点之后”同宿分叉系统受参激白噪声影响的分叉行为．     

随机扰动的同宿分岔系统研究

利用一维扩展过程的奇点理论并结合能量包络的随机平均法，考查“隐藏在余维２分岔点之后”的同宿分岔系统受参激白噪声影响的分岔行为。     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.