首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 31 毫秒
1.
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.  相似文献   

2.
Suppose an autonomous functional differential equation has an orbit Γ which is homoclinic to a hyperbolic equilibrium point. The purpose of this paper is to give a procedure for determining the behavior of the solutions near Γ of a functional differential equation which is a nonautonomous periodic perturbation of the original one. The procedure uses exponential dichotomies and the Fredholm alternative. It is also shown that any smooth function p(t) defined on the reals which approaches zero monotonically as t → ± ∞ is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero. Examples illustrating the results are also given.  相似文献   

3.
In this paper we discuss a small nonautonomous perturbation of an autonomous system on Rn which has a homoclinic solution. Regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co-existence of homoclinic orbits. Those conditions of various co-existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations.  相似文献   

4.
Behavior of the separatrix solution y ( t )=−(3/2)/cosh2( t /2) (homoclinic connection) of the second order equation y "= y + y 2 that undergoes the singular perturbation ɛ2 y ""+ y "= y + y 2, where ɛ>0 is a small parameter, is considered. This equation arises in the theory of traveling water waves in the presence of surface tension. It has been demonstrated both rigorously [1, 2] and using formal asymptotic arguments [3, 4] that the above-mentioned solution could not survive the perturbation.The latter papers were based on the Kruskal–Segur method (KS method), originally developed for the equation of crystal growth [5]. In fact, the key point of this method is the reduction of the original problem to the Stokes phenomenon of a certain parameterless "leading-order" equation. The main purpose of this article is further development of the KS method to study the breaking of homoclinic connections. In particular: (1) a rigorous basis for the KS method in the case of the above-mentioned perturbed problem is provided; and (2) it is demonstrated that breaking of a homoclinic connection is reducible to a monodromy problem for coalescing (as ɛ→0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.  相似文献   

5.
This paper considers the coupled KdV-type Boussinesq system with a small perturbation $u_{xx}=6cv-6u-6uv+\varepsilon f(\varepsilon,u,u_{x},v,v_{x}),$ $ v_{xx}=6cu-6v-3u^{2}+\varepsilon g(\varepsilon,u,u_{x},v,v_{x}),$ where $c=1+\mu$, $\mu>0$ and $\varepsilon$ are small parameters. The linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. We first change this system into an equivalent system with dimension 4, and then show that its dominant system has a homoclinic solution and the whole system has a periodic solution if the perturbation functions $g$ and $h$ satisfy some conditions. By using the contraction mapping theorem, the perturbation theorem, and the reversibility, we theoretically prove that this homoclinic solution, when higher order terms are added, will persist and exponentially approach to the obtained periodic solution (called generalized homoclinic solution) for small $\varepsilon$ and $\mu>0$.  相似文献   

6.
Rigorous Melnikov analysis is accomplished for Davey–Stewartson II equation under singular perturbation. Unstable fiber theorem and center-stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well posedness is necessary for such a Melnikov analysis. The main open issue regarding a proof of the existence of a homoclinic orbit to the perturbed Davey–Stewartson II equation is discussed in the Appendix .  相似文献   

7.
It is known that small perturbations of a Fredholm operator L have nulls of dimension not larger than dim N (L). In this paper for any given positive integer k≤dim N(L) we prove that there is a perturbation of L which has an exactly κ-dimensional null. Actually, our proof gives a construction of the perturbation. We further apply our result to concrete examples of differential equations with degenerate homoclinic orbits, showing how many independent homoclinic orbits can be bifurcated from a perturbation.  相似文献   

8.
The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.  相似文献   

9.
In this paper, the perturbation-incremental method is presented for the analysis of a quadratic isochronous system. This method combines the remarkable characteristics of the perturbation method and the incremental method. The first step is the perturbation method. Assume that the parameter $\lambda$ is small, i.e. $\lambda\approx0$, the initial expression of the homoclinic orbit is obtained. The second step is the parameter incremental method. By extending the solution corresponding to small parameters to large parameters, we can get the analytical-expressions of homoclinic orbits.  相似文献   

10.
Regarding the small perturbation as a parameter in an appropriate space of functions, we can discuss co-existence of homoclinic orbits for non-autonomous perturbations of an autonomous system in Rn and describe conditions of parameters for such degenerate homoclinic bifurcations with some bifurcation manifolds of infinite dimension. Since those manifolds determine the relation among parameters for such bifurcations, in this paper we give an algorithm to compute approximately those manifolds and concretely obtain their first order approximates.  相似文献   

11.
We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.  相似文献   

12.
Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.  相似文献   

13.
This paper deals with the numerical analysis of time dependent parabolic partial differential equation. The equation has bistable nonlinearity and models electrical activity in a neuron. A qualitative analysis of the model is performed by means of a singular perturbation theory. A small parameter is introduced in the highest order derivative term. This small parameter is known as singular perturbation parameter. Boundary layers occur in the solution of singularly perturbed problems when the singular perturbation parameter tend to zero. These boundary layers are located in neighbourhoods of the boundary of the domain, where the solution has a very steep gradient. Most of the conventional methods fails to capture this effect. A numerical scheme is constructed to overcome this discrepancy in literature. A rigorous analysis is carried out to obtain a-priori estimates on the solution of the problem and its derivatives. It is then proven that the numerical method is unconditionally stable. Convergence and stability analysis is carried out. A set of numerical experiment is carried out and it is observed that the scheme faithfully mimics the dynamics of the model.  相似文献   

14.
研究驻波广义Fisher-Kolmogorov方程u″″-βu″+u~3-u=0,β0.该方程有一个鞍中心型平衡点u=0(一对非零实特征值和一对纯虚特征值).应用扰动理论和调整相移,证明对每一个正常数β该方程在原点附近有一个连接周期解的同宿轨(该文称为广义同宿轨).  相似文献   

15.
We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.  相似文献   

16.
This paper is devoted to the analysis of the dynamic behavior of a three-species food chain model, in which two predators compete for the same prey while one of the predators feeds on the other. Under the assumption that the time responses of the three trophic levels are extremely diversified, the model is proved to have homoclinic orbit. We firstly use geometric singular perturbation method to detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Then, we show, numerically, that there exist also nonsingular homoclinic orbits that tend toward the singular ones for slightly different parameter values. This analysis is particularly helpful to understanding the chaotic behavior of the food chains.  相似文献   

17.
In the paper to be commented on, the authors may make some mistakes when reaching Eq. (15) by using the parameter perturbation method, i.e., perturbing about the degenerate point. This is highlighted here to avoid possible failure when the results are used in finding the parameter space where homoclinic bifurcation occurs. Furthermore, following the idea of the original paper, a feasible modified version of the original results is presented.  相似文献   

18.
We study homoclinic orbits to a saddle-center of a fourth-order ordinary differential equation, which is invariant under the transformation x→−x, involving an eigenvalue parameter q and an odd, piece-wise, cubic-type nonlinearity. It is found that for a sequence of eigenvalues which tends to infinity, homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger. These orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x→±∞.  相似文献   

19.
Summary The existence of homocliic orbits, for a finite-difference discretized form of a damped and driven perturbation of the focusing nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, for external parameters on a codimension 1 submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singular perturbation theory and a purely geometric argument (called the “second measurement” in the paper). The geometric singular perturbation theory deals with persistence of invariant manifolds and fibration of the persistent invariant manifolds. The approximate location of the codimension 1 submanifold of parameters is calculated. (This is the material in Part I.) Then, in a neighborhood of these homoclinic orbits, the existence of “Smale horseshoes” and the corresponding symbolic dynamics are established in Part II [21].  相似文献   

20.
A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg-Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg-Landau equation in terms of Fourier series.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号