Limit Cycles Bifurcating from a Quadratic Reversible Lotka-Volterra System with a Center and Three Saddles

This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits. The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.     

以抛物弓形为边界的周期环域的三次系统的Poincaré分支

本文以抛物弓形为边界的周期环域的三次系统的Poincaré分支为例,说明具有相同边界的周期环域的相同次数的多项式系统的Poincaré分支,由于周期环域内闭轨的不同,它们所对应的Abel积分也不同,所以它们的Poincaré分支所能分支出极限环的个数也是不同的.     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

The cyclicity of the period annulus of two classes of cubic isochronous systems

In this paper, we investigate the cyclicity of the period annulus of two classes of cubic isochronous systems.By using the Chebyshev criterion, we prove that the two systems have respectively at most three and four limit cycles produced fromthe period annulus around the isochronous center under cubic perturbations.     

The cyclicity of the period annulus of the cubic isochronous center

This paper is concerned with limit cycles which bifurcate from periodic orbits of the cubic isochronous center. It is proved that in this situation, the cyclicity of the period annulus under cubic perturbations is equal to four. Moreover, for each k?=?0,1, . . .,4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.     

On the period function of planar systems with unknown normalizers

A necessary and sufficient condition for the period function's monotonicity on a period annulus is given. The approach is based on the theory of normalizers, but is applicable without actually knowing a normalizer. Some applications to polynomial and Hamiltonian systems are presented.

     

Quadratic perturbations of a class of quadratic reversible center of genus one

This paper is concerned with the quadratic perturbations of a one-parameter family of quadratic reversible system, having a center of genus one. The exact upper bound of the number of limit cycles emerging from the period annulus surrounding the center of the unperturbed system is given.     

Changes in the mechanical properties of intervertebral lumbar disks produced by prolonged static compression forces

Conclusions These results indicate that the mechanical properties of intervertebral disks upon prolonged static compression undergo certain changes. The intradisk pressure in unaltered and moderately degenerated disks have a definite tendency to decrease and the dynamics of this process is related to the extent of degenerative changes in the disk. The changes in the intradisk pressure correlate with the changes in the tangential stretching of the annulus fibrosus. An overstretching effect was found for the annulus fibrosus in prolonged compression. The annulus fibrosus of the low lumbar disks which has a greater reserve of elasticity is more resistant to prolonged compression. In moderately degenerated disks, the turgor of the nucleus pulposus and elasticity of the annulus fibrosus are reduced and, thus, the reduction in the intradisk pressure and the extensibility of the annulus fibrosus upon the action of prolonged loads are not as pronounced. However, the changes found in these disks are very stable and are retained for a long period after the removal of compression. The intradisk pressure and tangential stretching of the annulus fibrosus at high loads remained virtually invariant during prolonged compression. This finding indicates that the structural elements of the disk under these conditions function at their limiting capacity. All the changes in the mechanical properties of the disks found are small, which indicates the high stability of human intervertebral lumbar disks relative to prolonged static compressive loads.Paper presented at the Second All-Union Conference on Biomechanics, Riga, April 1979.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1076–1080, November–December, 1979.     

Locating a minisum annulus: a new partial coverage distance model

The problem is to find the best location in the plane of a minisum annulus with fixed width using a partial coverage distance model. Using the concept of partial coverage distance, those demand points within the area of the annulus are served at no cost, while for ‘uncovered’ demand points there will be additional costs proportional to their distances to the annulus. The objective of the problem is to locate the annulus such that the sum of distances from the uncovered demand points to the annulus (covering area) is minimized. The distance is measured by the Euclidean norm. We discuss the case where the radius of the inner circle of the annulus is variable, and prove that at least two demand points must be on the boundary of any optimal annulus. An algorithm to solve the problem is derived based on this result.     

Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

Melnikov Functions for a Class of Piecewise Hamiltonian Systems

This paper is concerned with the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. By constructing a Poincar\''{e} map, we obtain explicit expressions of the first, second and third order Melnikov functions. In addition, we apply their expressions to give upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.     

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.     

A method for characterizing nilpotent centers

To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and second for studying the bifurcation of limit cycles from it or from its period annulus. We provide necessary conditions for detecting nilpotent centers based on recent developments. Moreover we survey the last results on this problem and illustrate our approach by means of examples.     

The period function of generalized Loudʼs centers

In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of Loud?s systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loud?s centers.     

The period function of reversible quadratic centers

In this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers, namely the dehomogenized Loud's systems. The local bifurcation diagram of the period function at the center is fully understood using the results of Chicone and Jacobs [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433-486]. Most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center. The techniques that we use here are different from the ones in [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433-486] because, while the period function extends analytically at the center, it has no smooth extension to the polycycle. At best one can hope that it has some asymptotic expansion. Another major difficulty is that the asymptotic development has to be uniform with respect to the parameters, in order to prove that a parameter is not a bifurcation value. We study also the bifurcations in the interior of the period annulus and we show that there exist three germs of curves in the parameter space that correspond to this type of bifurcation. Moreover we determine some regions in the parameter space for which the corresponding period function has at least one or two critical periods. Finally we propose a complete conjectural bifurcation diagram of the period function of the dehomogenized Loud's systems. Our results can also be viewed as a contribution to the proof of Chicone's conjecture [C. Chicone, review in MathSciNet, ref. 94h:58072].     

Offset-polygon annulus placement problems

An offset-polygon annulus region is defined in terms of a polygon P and a distance δ > 0 (offset of P). In this paper we solve several containment problems for polygon annulus regions with respect to an input point set. Optimization criteria include both maximizing the number of points contained in a fixed size annulus and minimizing the size of the annulus needed to contain all points. We address the following variants of the problem: placement of an annulus of a convex polygon as well as of a simple polygon; placement by translation only, or by translation and rotation; off-line and on-line versions of the corresponding decision problems; and decision as well as optimization versions of the problems. We present efficient algorithms in each case.     

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x+bx3-x5+ε(α+βx2+γx4)y,where b∈R,0|ε|1,(α,β,γ)∈D∈R3 and D is bounded.We prove that for |b|1(b0) the least upper bound of the number of isolated zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx is 2(counting the multiplicity) and this upper bound is a sharp one.     

两连域保角映射成环域的方法

本文论述一种把型区域保角映射成环域的方法.其要点是将问题转化为Dirichlet问题,并证明该映像函数之实部应满足本文所示的边界条件,进而依据两连域上定义的调和函数的单值特性确定环域的内半径.映像函数的虚部可由Cauchy-Riemann条件得到,由此产生的积分常数仅影响映像点的幅角,并可由一一对应的映像来确定.不失其一般性,本方法可将由矩形拼成的复杂两连域保角映射成环域.笔者还对本方法作了电算,证明本方法可靠、经济、结果附有表格.     

The cyclicity and period function of a class of quadratic reversible Lotka-Volterra system of genus one

In this paper, we investigate a class of quadratic reversible Lotka-Volterra system of genus one with b=3/5. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. Moreover, we prove that the period function of its period trajectories is monotone increasing.     

Remarks on minimal annuli in a slab

We show that the two closed boundary curves of a minimal annulus in a slab are both convex if one of them is convex and along the other curve the surface meets the plane at a constant angle. And therefore, under the same condition, the minimal annulus is foliated by convex planar curves all of which are parallel to the boundary. In particular, if the convex curve is a circle, then the annulus is part of a catenoid.