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1.
具有一类三次曲线解的Kolmogorov三次系统的极限环的存在性 总被引:1,自引:0,他引:1
证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的. 相似文献
2.
WANGXIANGRONG HANMAOAN 《高校应用数学学报(英文版)》1998,13(4):385-390
The authors consider the nonhnear systems x=h(y)-F(x),y=-g(x) in which g(x) may be not differentiabte and the system can be nonsymmetric. Some conditions which en-sure that there exists an infinite number of limit cycles are obtained. A problem about center-focus put forward by R. Conti has been answered. 相似文献
3.
We present some properties of a differential system that can be used to model intratrophic predation in simple predator-prey models. In particular, for the model we determine the maximum number of limit cycles that can exist around the only fine focus in the first quadrant and show that this critical point cannot be a centre. 相似文献
4.
Xuncheng Huang Yuanming Wang Lemin Zhu 《Mathematical Methods in the Applied Sciences》2007,30(5):501-511
A cubic differential system is proposed, which can be considered a generalization of the predator–prey models, studied recently by many authors. The properties of the equilibrium points, the existence of a uniqueness limit cycle, and the conditions for three limit cycles are investigated. The criterion is easy to apply in applications. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
5.
We present necessary and sufficient conditions for a critical point of certain two-dimensional cubic differential systems to be a centre. Extensive use of the computer algebra system REDUCE is involved. The search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations. It has proved to be a difficult problem, and full conditions are known for very few classes of systems. Such conditions are also required in the investigation of Hilbert's sixteenth problem concerning the number of limit cycles of polynomial systems. 相似文献
6.
7.
Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields. 相似文献
8.
主要研究一类三次系统的极限环存在性问题,推广了C.Chicone[2]的结果,给出此类系统极限环存在定理. 相似文献
9.
在混合扰动下从闭轨族分支的极限环 总被引:1,自引:0,他引:1
本文讨论了对确定微分方程组的向量场和分析其轨线穿过方向的参考闭曲线族同时进行扰动分支极限环的方法,并给出了一个平面二次微分系统在混合扰动下分支出三个极限环的例子 相似文献
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11.
In this paper, we give some evidences what cause more limit cycles for piecewise dynamical systems. We say, the angles or the number of zones are critical points. We study an example of linear lateral systems and an example of linear Y-shape systems, and prove that they have five and four crossing limit cycles by using Newton-Kantorovich Theorem, respectively. 相似文献
12.
This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable. 相似文献
13.
Jaume Llibre Manuel Ordóñez Enrique Ponce 《Nonlinear Analysis: Real World Applications》2013,14(5):2002-2012
Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems.New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.A relevant application to the McKean piecewise linear model of a single neuron activity is included. 相似文献
14.
We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles. 相似文献
15.
A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities
This paper is devoted to study the planar polynomial system: where and are homogeneous polynomials of degree . Denote . We prove that the system has at most 1 limit cycle surrounding the origin provided . Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin. 相似文献
16.
Theory of center-focus for a class of higher-degree critical points and infinite points 总被引:1,自引:0,他引:1
LIU Yirong 《中国科学A辑(英文版)》2001,44(3):365-377
For the real planar autonomous differential system, the questions of detection between center and focus, successor function,
formal series, central integration, integration factor, focal values, values of singular point and bifurcation of limit cycles
for a class of higher-degree critical points and infinite points are expounded. 相似文献
17.
Yusen Wu Peiluan LiHaibo Chen 《Communications in Nonlinear Science & Numerical Simulation》2012,17(1):292-304
In the present paper, for the three-order nilpotent critical point of a cubic Lyapunov system, the center problem and bifurcation of limit cycles are investigated. With the help of computer algebra system-MATHEMATICA, the first 7 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist 7 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for cubic Lyapunov systems. 相似文献
18.
E. Sáez L Szántó 《数学物理学报(B辑英文版)》2008,28(4):865-869
In this article,the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola.For this class of second-order differential systems,it is shown that for certain values of the parameters the invariant parabola coexists with a center.For other values it can coexist with one,two or three small amplitude limit cycles which are constructed by Hopf bifurcation.This result gives an answer for the question given in[4],about the existence of limit cycles for such class of system. 相似文献
19.
Vimal Singh 《Journal of Difference Equations and Applications》2013,19(12):1255-1265
Using Lyapunov's direct method, a novel frequency-domain criterion for the elimination of limit cycles in a class of digital filters using single saturation nonlinearity is derived. The criterion turns out to be a generalization and improvement over an earlier criterion due to Kar and Singh. An example showing the effectiveness of the criterion is given. A graphical interpretation of a simplified version (involving one free parameter) of the criterion is discussed. 相似文献
20.
讨论了一类三次系统x=-y(1-βx2)-(a1x a2x2 a3x3),y=b1x b2x2 b3x3的极限环问题.对包含一个奇点或多个奇点的极限环的唯一性和唯二性给出了若干充分条件. 相似文献