共查询到20条相似文献,搜索用时 31 毫秒
1.
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. 相似文献
2.
Yang Junmin Han Maoan 《Annals of Differential Equations》2007,23(4):536-545
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. 相似文献
3.
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. 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. 相似文献
4.
For a piecewise analytical Hamiltonian system with a cusp on a switch line, which has a family of periodic orbits near a generalized homoclinic loop, we study the maximum number of limit cycles bifurcating from the periodic orbits. For doing so, we first obtain the asymptotic expressions of the Melnikov functions near the loop. Finally we present two examples illustrating applications of the main results. 相似文献
5.
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 相似文献
6.
HOMOCLINIC BIFURCATION WITH CODIMENSION 3 总被引:5,自引:0,他引:5
Zhu Deming 《数学年刊B辑(英文版)》1994,15(2):205-216
HOMOCLINICBIFURCATIONWITHCODIMENSION3¥ZHUDEMINGAbstract:FirstitisprovedthatboththeintegralofthedivergenceandtheMelnikovfuncti... 相似文献
7.
On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation 下载免费PDF全文
Jihua Yang 《Journal of Applied Analysis & Computation》2020,10(6):2362-2380
This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated. 相似文献
8.
Consideraplanarperturbedsystemoftheformx=Hy+εf(x,y),y=-Hx+εg(x,y)(1)whereH,f,garefunctionsofclasC∞.Asumeforε=0(1)tohaveahomoc... 相似文献
9.
We concern the number of limit cycles of a polynomial system with degree nine. We prove that under different conditions, the system can have 12 and 20 limit cycles bifurcating from a compound loop with five saddles. Our method relies upon the Melnikov function method and the method of stability-changing of a double homoclinic loop proposed by the authors[J. Yang, Y. Xiong and M. Han, {\em Nonlinear Anal-Theor.}, 2014, 95, 756--773]. 相似文献
10.
Melnikov method and detection of chaos for non-smooth systems 总被引:1,自引:0,他引:1
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. 相似文献
11.
Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results. 相似文献
12.
In this paper, we deal with the problem of limit cycle bifurcation near a 2-polycycle or 3-polycycle for a class of integrable systems by using the first order Melnikov function. We first get the formal expansion of the Melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion. Based on the coefficients, we obtain a lower bound for the maximal number of limit cycles near the polycycle. As an application of our main results, we consider quadratic integrable polynomial systems, obtaining at least two limit cycles. 相似文献
13.
In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer , we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for , we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles. 相似文献
14.
In this paper, we first study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find 4, 5 or 6 limit cycles. 相似文献
15.
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. 相似文献
16.
Josefina CASASAYAS Patrícia FAISCA Ana NUNES 《NoDEA : Nonlinear Differential Equations and Applications》2003,10(1):119-131
17.
Xianbo SunMaoan Han Junmin Yang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(9):2948-2965
In this article, we study the expansion of the first Melnikov function of a near-Hamiltonian system near a heteroclinic loop with a cusp and a saddle or two cusps, obtaining formulas to compute the first coefficients of the expansion. Then we use the results to study the problem of limit cycle bifurcation for two polynomial systems. 相似文献
18.
Ali Atabaigi Hamid R.Z. Zangeneh Rasool Kazemi 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):1945-1958
This paper deals with the analytical property of the first Melnikov function for general Hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the Hamiltonian value corresponding to the loop. The explicit formulas for the first coefficients of the expansion have been given. We prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions. Then we consider the cyclicity of a cuspidal loop in some Liénard and Hamiltonian systems, and determine the number of limit cycles that can bifurcate from the perturbed system. 相似文献
19.
Chaotic response of a quarter car model forced by a road profile with a stochastic component 总被引:1,自引:0,他引:1
Grzegorz Litak Marek Borowiec Michael I. Friswell Wojciech Przystupa 《Chaos, solitons, and fractals》2009,39(5):2448-2456
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. 相似文献
20.
《Communications in Nonlinear Science & Numerical Simulation》2008,13(7):1373-1383
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. 相似文献