On the period function of reversible quadratic centers with their orbits inside quartics

This paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics. It is proved that such a system has a period function with at most one critical point.     

On the reversible quadratic centers with monotonic period function

This paper is devoted to studying the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

We determine several regions in the parameter plane for which the corresponding center has a monotonic period function. To this end we first show that any of these systems can be brought by means of a coordinate transformation to a potential system. Then we apply a monotonicity criterium of R. Schaaf.

     

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].     

On the monotonicity of the period function of reversible centers On the monotonicity of the period function of reversible centers

In this paper we study the period function of centers for a class of reversible systems and give a criteria to determine the monotonicity of the period functions.     

The period function for quadratic integrable systems with cubic orbits

This paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves. It is proved that each such system has a period function with at most one critical point.     

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.     

On the period function in a class of generalized Lotka-Volterra systems

In this note, motivated by the recent results of Wang et al. [Wang et al., Local bifurcations of critical periods in a generalized 2D LV system, Appl. Math. Comput. 214 (2009) 17-25], we study the behaviour of the period function of the center at the point (1,1) of the planar differential system

     

The period function of the generalized Lotka-Volterra centers

The present paper deals with the period function of the quadratic centers. In the literature different terminologies are used to classify these centers, but essentially there are four families: Hamiltonian, reversible , codimension four Q₄ and generalized Lotka-Volterra systems . Chicone [C. Chicone, Review in MathSciNet, Ref. 94h:58072] conjectured that the reversible centers have at most two critical periods, and that the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, only the monotonicity of the Hamiltonian and Q₄ families [W.A. Coppel, L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993) 1357-1365; Y. Zhao, The monotonicity of period function for codimension four quadratic system Q₄, J. Differential Equations 185 (2002) 370-387] has been proved. Concerning the family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka-Volterra centers [F. Rothe, The periods of the Volterra-Lokta system, J. Reine Angew. Math. 355 (1985) 129-138; R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984) 1-31; J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986) 178-184]. By means of the first period constant one can easily conclude that the period function of the centers in the family is monotone increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone's conjecture, it should be also monotone increasing near the outer boundary, which in the Poincaré disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there is no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.     

关于一类二次哈密尔顿系统在二次扰动下的极限环个数问题

本文研究了一类具有两个鞍点和一个中心的通用二次哈密尔顿向量场在二次扰动下的三参数开折，证明极限环的最小上界为２。     

On a class of quadratic programs

This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.     

A criticality result for polycycles in a family of quadratic reversible centers

We consider the family of dehomogenized Loud's centers $X_{\mu }=y(x?1){?}_x+(x+D{x}^2+F{y}^2){?}_y$, where $\mu =(D,F)\in {\mathbb{R}}^2$, and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu },\mu \in {\mathbb{R}}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ${\mathrm{\Gamma }}_B$ of codimension 1 in ${\mathbb{R}}^2$. In the present paper we succeed in proving that a subset of ${\mathrm{\Gamma }}_B$ has criticality equal to one.     

Quadratic perturbations of quadratic codimension-four centers

We study the stratum in the set of all quadratic differential systems , with a center, known as the codimension-four case Q₄. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I. We show that the orbits of the unperturbed system are elliptic curves, and I is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.     

The period function of potential systems of polynomials with real zeros

We consider some analytic behaviors (convexity, monotonicity and number of critical points) of the period function of period annuli of the potential system and focus on the case when g(x) is a polynomial whose roots are all real. The main contributions of this paper are twofold: (i) analytic behaviors are given for the period functions of period annuli surrounding one or more and simple or degenerate equilibria; (ii) as a nontrivial application of the general conclusions in (i), a purely analytical and shorter proof is provided for a result for the case degg=4 recently obtained by Chengzhi Li and Kening Lu with some help of computer algebra [Chengzhi Li, Kening Lu, The period function of hyperelliptic Hamiltonian of degree 5 with real critical points, Nonlinearity 21 (2008) 465-483].     

On real quadratic function fields of Chowla type with ideal class number one

Let be the finite field with elements, (2), , where is a square-free polynomial in with and . In this paper several equivalent conditions for the ideal class number to be one are presented and all such quadratic function fields with are determined.

     

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x^″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n^-1|y|ⁿ. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γu^γ-1 in two dimensions.     

On the period function of x″+f(x)x′+g(x)=0

We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.     

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for their ideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic function fields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal class numbers of these function fields are divisible by n.     

Quadratic perturbations of a quadratic reversible center of genus one

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al.