高间Melnikov方法

本文把原有Melnikov方法推广到高阶情况.找到了二阶次谐Melnikov函数表达式,并且证明了在一定条件下可以用二阶次谐Melnikov函数来判定系统的次谐或超次谐的存在.     

Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

Chaotic Behavior and Subharmonic Bifurcations for the Duffing-van Der Pol Oscillator

Chaotic behavior for the Duffing-van der Pol (DVP) oscillator is investigated both analytically and numerically. The critical curves separating the chaotic and non-chaotic regions are obtained. The chaotic feature on the system parameters are discussed in detail. The conditions for subharmonic bifurcations are also obtained. Numerical results are given, which verify the analytical ones.     

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers e ^±iϕ . On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.      

一类扩张了的软弹簧型Duffing方程的紊动性态

本文用Melnikov函数方法讨论了一类扩张了的软弹簧型Duffing方程(k=1,2,3,…)在周期激励下的紊动现象.给出了出现二阶同宿切的条件.文中所采用的方法对于不能给出并宿轨道的显式的系统的研究是非常有用的.     

Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-Ⅳ Functional Response

In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳfunctionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.     

Melnikov method and detection of chaos for non-smooth systems

We extend the Melnikov method to non-smooth dynamical systems to study the global behavior near a non-smooth homoclinic orbit under small time-periodic perturbations. The definition and an explicit expression for the extended Melnikov function are given and applied to determine the appearance of transversal homoclinic orbits and chaos. In addition to the standard integral part, the extended Melnikov function contains an extra term which reflects the change of the vector field at the discontinuity. An example is discussed to illustrate the results.     

HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

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Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

Global bifurcations and single‐pulse homoclinic orbits of a plate subjected to the transverse and in‐plane excitations

The Shilnikov‐type single‐pulse homoclinic orbits and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in‐plane excitations are investigated in detail. The resonant case considered here is principal parametric resonance and 1:2 internal resonance. Based on the normal form theory, the desired form for the global perturbation method is obtained. By using the global perturbation method developed by Kovacic and Wiggins, explicit sufficient conditions for the existence of a Shilnikov‐type homoclinic orbit are obtained, which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes. Numerical results obtained by using the fourth‐order Runge–Kutta method agree with theoretical analysis at least qualitatively. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiple Bifurcations in a Polynomial Model of Bursting Oscillations

Summary. Bursting oscillations are commonly seen to be the primary mode of electrical behaviour in a variety of nerve and endocrine cells, and have also been observed in some biochemical and chemical systems. There are many models of bursting. This paper addresses the issue of being able to predict the type of bursting oscillation that can be produced by a model. A simplified model capable of exhibiting a wide variety of bursting oscillations is examined. By considering the codimension-2 bifurcations associated with Hopf, homoclinic, and saddle-node of periodics bifurcations, a bifurcation map in two-dimensional parameter space is created. Each region on the map is characterized by a qualitatively distinct bifurcation diagram and, hence, represents one type of bursting oscillation. The map elucidates the relationship between the various types of bursting oscillations. In addition, the map provides a different and broader view of the current classification scheme of bursting oscillations. Received October 9, 1996; revised June 16, 1997, and accepted September 26, 1997     

双曲极限环在周期扰动下次调和解的分支

研究了一给定平面自治系统的双曲极限环在周期扰动下m阶次调和解的分支问题,用Poincar啨映射,通过变尺度方法,获得了判别m阶次调和解的存在条件,最后给出了一个实例。     

非线性振动系统的异宿轨道分叉,次谐分叉和混沌

在参数激励与强迫激励联合作用下具有van der Pol阻尼的非线性振动系统,其动态行为是非常复杂的.本文利用Melnikov方法研究了这类系统的异宿轨道分叉、次谐分叉和混沌.对于各种不同的共振情况,系统将经过无限次奇阶次谐分叉产生Smale马蹄而进入混沌状态.最后我们利用数值计算方法研究了这类系统的混沌运动.所得结果揭示了一些新的现象.     

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．     

Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting

This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie~Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14（1） （2010）, 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.     

非线性弹性梁中的混沌带现象

研究了非线性弹性梁的混沌运动,梁受到轴向载荷的作用。非线性弹性梁的本构方程可用三次多项式表示。计及材料非线性和几何非线性,建立了系统的非线性控制方程。利用非线性Galerkin法,得到微分动力系统。采用Melnikov方法对系统进行分析后发现,当载荷P₀和f满足一定条件时,系统将发生混沌运动,且混沌运动的区域呈现带状。还详尽分析了从次谐分岔到混沌的路径,确定了混沌发生的临界条件。     

On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies

We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings.      

周期时间制约的非Hamliton可积Kolmogorov系统的混沌与次谐波

本文研究一类非Hamilton可积的Kolmogorov生态系统的周期激励模型，应用Melnikov方法，得到了该系统存在混沌与次谐分枝的某些充分条件。     

Melnikov方法和圆型平面限制性三体问题的横截同宿研究*

本文对由两自由度近可积哈密顿系统经非正则变换而得到的,具有高阶不动点的非哈密顿系统给出了判别横截同宿轨和横截异宿轨存在性的两条判据。对原二体质量比很小时近可积圆型平面限制性三体问题,采用本文判据证明存在横截同宿轨,从而存在横截同宿穿插现象;还在一定假设下证明了存在横截异宿轨;并给出了全局定性相图。     

Bifurcations and chaos in Duffing equation with damping and external excitations

Duffng equation with damping and external excitations is investigated.By using Melnikov method and bifurcation theory,the criterions of existence of chaos under periodic perturbations are obtained.By using second-order averaging method,the criterions of existence of chaos in averaged system under quasi-periodic perturbations forff=nω+εσ,n=2,4,6(whereσis not rational toω)are investigated.However,the criterions of existence of chaos for n=1,3,5,7-20 can not be given.The numerical simulations verify the theoretical analysis,show the occurrence of chaos in the averaged system and original system under quasiperiodic perturbation for n=1,2,3,5,and expose some new complex dynamical behaviors which can not be given by theoretical analysis.In particular,the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations,and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.