Cyclicity of a kind of degenerate polycycles through three singular points

This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node P0 and two hyperbolic saddles P1 and P2, where the hyperbolicity ratio of the saddle P1 (which connects the saddle-node with hh-connection) is equal to 1 and that of the other saddle P2 is irrational. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. Then the cyclicity of this kind of polycycle is no more than m 3 if the saddle P1 is of order m and the hyperbolicity ratio of P2 is bigger than m. Furthermore, the cyclicity of this polycycle is no more than 7 if the saddle P1 is of order 2 and the hyperbolicity ratio of P2 is located in the interval (1,2).     

Bifurcation phenomenon of a class of planar codimension 3 polycycleS (2) with two saddles resonating

By means of finitely-smooth normal form theory and the method of infinitesimal analysis, it is proved that the cyclicity of a planar codimension 3 polycycleS(2) is three and the complete bifurcation diagram is provided. Project supported by the National Natural Science Foundation of China and the DEPF of China.     

平面可积系统的多角环S^（2）在小扰动下的环性

本文应用有限光滑正规形理论,对可积系统的多角环Ｓ＾（２）,在小扰动下,当满足一定的非退化条件时,证明其环性为２。     

Cyclicity of planar polycycles and ensembles with codimension 3 degeneration through a saddle-node and two hyperbolic saddles

The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.     

Cyclicity of Degenerate Polycycles Through a Saddle-Node and Two Hyperbolic Saddles

This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle （which connects the saddle-node with hp-connection） is equal to 1 and that of the other saddle is irrational. It is obtained that the cyclicity of this kind of polycycle is no more than 5 if the hp-connection keeps unbroken under the C^∞ perturbations.     

Multiple limit cycles bifurcation from the degenerate singularity for a class of three-dimensional systems

In this paper, bifurcation of small amplitude limit cycles from the degenerate equilibrium of a three-dimensional system is investigated. Firstly, the method to calculate the focal values at nilpotent critical point on center manifold is discussed. Then an example is studied, by computing the quasi-Lyapunov constants, the existence of at least 4 limit cycles on the center manifold is proved. In terms of degenerate singularity in high-dimensional systems, our work is new.     

Generating limit cycles from a nilpotent critical point via normal forms

It is well known that the normal form theory can be applied to solve the center-focus problem for monodromic planar nilpotent singularities. In this paper we see how this theory can also be applied to generate limit cycles from this type of singularities.     

On the structure of the set bases of a degenerate point

Consider an extreme point (EP)x ⁰ of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox ⁰ is degenerate,x ⁰ is called a degenerate EP. Corresponding tox ⁰, there are several bases. We will characterize the set of all bases associated withx ⁰, denoted byB ⁰. The setB ⁰ can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB ⁰, there is a corresponding undirected graphG ⁰, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG ⁰ and cycling (e.g., in linear programs).     

Linear estimate of the number of limit cycles for a class of non-linear systems

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple zeros, which also gives the upper bound of the number of limit cycles for the given system.     

Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology

In this work we consider the reaction‐diffusion system of FitzHugh‐Nagumo type describing the behavior of the electrical conduction in an anisotropic cardiac muscle. The analysis of the Galerkin semidiscrete space approximation to this system is approached by means of a suitable variational formulation in the framework of abstract degenerate evolution equations. The main results concern convergence analysis and a priori stability estimates for the semidiscrete solution. These abstract results are then applied to the cardiac problem and for the finite element Galerkin approximation we achieve optimal order convergence. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 218–240, 2002; DOI 10.1002/num.1000     

一类2n+1次多项式微分系统的局部极限环分支

研究了一类2n 1次多项式微分系统在原点的局部极限环分支问题,通过计算与理论推导得出了该系统原点的奇点量表达式,确定了系统原点的中心条件以及最高阶细焦点的条件,并在此基础上构造出系统在原点分支出4个极限环的实例.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation , after a small polynomial perturbation. We first show that, under small perturbations of the form , where is a polynomial of degree 2m−1 in which the power of z is odd and the power of is even, the only possible distribution of limit cycles is (u,u) for all values of u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of is m−3, for m≥4. Then we consider a perturbation of the form , where is a polynomial of degree m in which the power of z is odd and obtain the upper bound m−5, for m≥6. Moreover, we show that the distribution (u,v) of limit cycles is possible for 0≤u≤m−5, 0≤v≤m−5 with u+v≤m−2 and m≥9.     

EXISTENCE，UNIQUENESS AND PROPERTIES OF THE SOLUTIONS OF A DEGENERATE PARABOLIC EQUATION WITH DIFFUSION－ADVECTION－ABSORPTION

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Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

On a class of degenerate elliptic operators arising from Fleming-Viot processes

We are dealing with the solvability of an elliptic problem related to a class of degenerate second order operators which arise from the theory of Fleming-Viot processes in population genetics. In the one dimensional case the problem is solved in the space of continuous functions. In higher dimension we study the problem in spaces with respect to an explicit measure which, under suitable assumptions, can be taken invariant and symmetrizing for the operators. We prove the existence and uniqueness of weak solutions and we show that the closure of the operator in such spaces generates an analytic -semigroup. Received December 4, 2000; accepted December 9, 2000.     

Birth of limit cycles bifurcating from a nonsmooth center

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ  -equivalent to a particular normal form _Z0$Z_0$. Given a positive integer number k   we explicitly construct families of piecewise smooth vector fields emerging from _Z0$Z_0$ that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ  -center of _Z0$Z_0$ (the same holds for k=∞$k=\infty$). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k   emerging from _Z0$Z_0$. As a consequence we prove that _Z0$Z_0$ has infinite codimension.     

Conditions for the existence of limit cycles of the second kind for a system of differential equations: II

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.