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1.
Yirong Liu 《Bulletin des Sciences Mathématiques》2004,128(2):77-89
In this article, the center conditions and isochronous center conditions at infinity for differential systems are investigated. We give a transformation by which infinity can be transferred into the origin. So we can study the properties of infinity with the methods of the origin. As an application of our method, we discuss the conditions of infinity to be a center and a isochronous center for a class of rational differential system. As far as we know, this is the first time that the isochronous center conditions of infinity are discussed. 相似文献
2.
For a class of cubic systems, the authors give a representation of
the $n${\rm th} order Liapunov constant through a chain of
pseudo-divisions. As an application, the center problem and the
isochronous center problem of a particular system are considered.
They show that the system has a center at the origin if and only if
the first seven Liapunov constants vanish, and cannot have an
isochronous center at the origin. 相似文献
3.
Xingwu Chen Valery G. Romanovski Weinian Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2008
In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin. 相似文献
4.
In this paper, we investigate a generalized two-dimensional Lotka-Volterra system which has a center. We give an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order. We show that with appropriate perturbations at most two critical periods bifurcate from the center. 相似文献
5.
In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained. 相似文献
6.
In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasianalytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than those of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system. 相似文献
7.
Jaume Gine Jaume Llibre Claudia Valls 《Journal of Applied Analysis & Computation》2017,7(4):1534-1548
For the polynomial differential system $\dot{x}=-y$, $\dot{y}=x +Q_n(x,y)$, where $Q_n(x,y)$ is a homogeneous polynomial of degree $n$ there are the following two conjectures done in 1999. (1) Is it true that the previous system for $n \ge 2$ has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all $n$ even. More precisely, we prove both conjectures in the case $n = 4$ and for $n\ge 6$ even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of $n$ odd was studied in [8]. 相似文献
8.
In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system. 相似文献
9.
Consider a family of planar systems having a center at the origin and assume that for ε=0 they have an isochronous center. Firstly, we give an explicit formula for the first order term in ε of the derivative of the period function. We apply this formula to prove that, up to first order in ε, at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient conditions for the existence of this critical period are explicitly given. From the tools developed in this paper we also provide a new characterization of planar isochronous centers. 相似文献
10.
In this paper, the conditions of center and isochronous center at the origin for a class of planar quartic differential systems are studied. At first, a constructive theorem of singular point quantities is presented, which plays an important role in simplifying periodic constants. The sufficient and necessary conditions for the origin of the systems being a center are obtained. Then a complete classification of the sufficient and necessary conditions are given for the origin of the systems being an isochronous center. 相似文献
11.
Xingwu Chen Wentao Huang Valery G. Romanovski Weinian Zhang 《Journal of Mathematical Analysis and Applications》2011,383(1):179-189
In this paper we study the linearizability problem of polynomial-like complex differential systems. We give a reduction of linearizability problem of such non-polynomial systems to the problem of polynomial systems. Applying this reduction, we find some linearizability conditions for a time-reversible quartic-like complex system and derive from them conditions of isochronous center for the corresponding real system. 相似文献
12.
13.
Yirong Liu 《Bulletin des Sciences Mathématiques》2003,127(2):133-148
The computation of period constants is a way to study isochronous center for polynomial differential systems. In this article, a new method to compute period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As an application, we discuss the center conditions and isochronous centers for a class of high-degree system. 相似文献
14.
Li-jun YangDepartment of Mathematical Sciences of Tsinghua University Beijing China 《应用数学学报(英文版)》2002,18(2):315-324
Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous. 相似文献
15.
研究了一类五次系统原点复等时中心的问题.先通过一种最新算法求出了这类五次系统原点的周期常数,从而得到复等时中心的必要条件,并利用一些有效途径证明它们的充分性.这实际上解决了这类五次系统的伴随系统原点等时中心问题与其自身为实系统时鞍点可线性化的问题. 相似文献
16.
研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性. 相似文献
17.
In this paper, we answer the question: under what conditions a class of rigid differential systems have a composition center. We give the sufficient and necessary conditions for these systems to have a center at origin point. At the same time, we give the formula of focal values and the highest order of fine focus. 相似文献
18.
Qin-long Wang Yi-rong Liu 《应用数学学报(英文版)》2007,23(3):451-466
In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given. 相似文献
19.
On the basis of some works on persistent centers and weakly persistent centers, in this paper we discuss a generalized version of persistent center and weakly persistent center for complex planar differential systems, in which conjugacy of variables may not be required. We give some complex systems which have a persistent center or weakly persistent center at the origin. Then, we find all conditions of persistent center for cubic systems and all conditions of weakly persistent center for complex cubic Lotka–Volterra system. Relations between complex systems and real ones are given concerning persistent centers and weakly persistent centers. 相似文献
20.
E. P. Volokitin 《Journal of Applied and Industrial Mathematics》2009,3(3):401-408
A method is proposed for deriving center conditions for uniformly isochronous systems of a particular form. The method is
based on reducing a system to the Abel ordinary differential equation. 相似文献