Melnikov method for discontinuous planar systems

This paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right-hand side. More precisely, we deal with a T$T$-periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit. By means of the so-called “non-smooth” Melnikov function there is shown the existence of a homoclinic solution for a perturbed system. The non-smooth Melnikov function is derived, and the method of how to find it in concrete problems is also introduced.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

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。     

On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems

The first-order Melnikov function of a homoclinic loop through a nilpotent saddle for general planar near-Hamiltonian systems is considered. The asymptotic expansion of this Melnikov function and formulas for its first coefficients are given. The number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. An example is presented as an application of the main results.     

奇摄动下的共振不变环面分支

本文研究高维退化系统在小扰动下的动力学行为,在共振的情况下,利用延拓的方法,讨论了扰动 系统不变环面的保存性,并利用推广的Melnikov函数、横截性理论讨论了同宿于不变环面的横截同宿 轨道存在的条件,推广和改进了一些文献的结果.     

指数二分性与高阶Melnikov函数

本文利用指数二分性理论和Liapunov－Schmidt方法，研究了当Melnikov函数具有高阶零点时的横截同宿轨道的存在性，得到了一个所谓的高阶Melnikov函数     

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。     

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.     

LIMIT CYCLES NEAR A DOUBLE HOMOCLINIC LOOP

In this article,we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop.We obtain an explicit formula to compute the first four coeffcients,and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients.Finally,we present some interesting applications.     

Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals

Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $W^H$.     

ON THE COEFFICIENTS APPEARING IN THE EXPANSION OF MELNIKOV FUNCTIONS IN HOMOCLINIC BIFURCATIONS

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Chaotic vibration of a quarter-car model excited by the road surface profile

The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by the road surface profile. By analyzing the potential an analytic expression is found for the homoclinic orbit. By introducing an harmonic excitation term and damping as perturbations, the critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically.     

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．     

Homoclinic Orbits for a Perturbed Lattice Modified KdV Equation

We establish the splitting of homoclinic orbits for a near-integrable lattice modified KdV (mKdV) equation with periodic boundary conditions. We use the Bäcklund transformation to construct homoclinic orbits of the lattice mKdV equation. We build the Melnikov function with the gradient of the invariant defined through the discrete Floquet discriminant evaluated at critical points. The criteria for the persistence of homoclinic solutions of the perturbed lattice mKdV equation are established.     

Analysis of the transient behavior in a two dof nonlinear system

In this paper we analyze the behavior of a nonlinear system under impulse loadings. The system is composed of a master “linear” degree of freedom (dof) substructure which is attached to a slave “nonlinear” energy sink (NES) for the sake of the control. Melnikov integral is endowed in order to study the possibility of existence of chaos and transversal homoclinic orbits in the system. Then, the complexification method as an alternative to nonlinear normal modes is implemented to reveal the behavior of the system during the energy exchange between two oscillators. The non-smooth time transformation (NSTT) technique is implemented in order to enlighten the system behavior during its extremely nonlinear regime, meanwhile stable and unstable zones of the system during its quasi-linear regime are highlighted.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.     

Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

Summary. We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2 ⁿ critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse function, in different kinds of examples. Received January 18, 1999; final revision received October 25, 1999; accepted December 12, 1999     

Exponential asymptotics of oscillatory integrals. Application to the problem of persistence of homoclinic connections for the 02iω resonance

We explain in this Note how to obtain an exponentially small equivalent of an oscillatory integral when it involves solutions of nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections to 0 for vector fields admitting a 0²iω resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small oscillatory integral.     

Chaotic response of a quarter car model forced by a road profile with a stochastic component

The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by a road surface profile consisting of harmonic and noisy components. By analyzing the potential an analytic expression is found for the homoclinic orbit. The road profile excitation including harmonic and random characteristics as well as the damping are treated as perturbations of a Hamiltonian system. The critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically. This transition is analyzed for different levels of noise and illustrated by numerical simulations.