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.     

长周期光纤光栅应变和温度传感灵敏度研究

基于模耦合理论,从长周期光纤光栅在外加轴向应变或环境温度变化时可能出现的弹性形变、弹光效应、热光效应、热膨胀效应及波导效应入手,推导了描述其轴向应变和温度灵敏度的一般性公式.利用数值方法研究了光栅周期、耦合模式的阶次、弹光系数及热光系数对灵敏度的影响.结果表明：在相同条件下,增大光栅周期将是提高灵敏度的有效手段之一;低次模较高次模的耦合具有更高的灵敏度;包层与芯区的弹光系数（热光系数）相等时,波长漂移量很小且与谐振波长值有简单的正比关系.     

Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y²+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.     

微扰模型分析长周期光纤光栅薄膜传感器

提出了一种新的基于三层阶跃折射率光纤的微扰模型,分析长周期光纤光栅（Long Period Fiber Grating,LPFG）薄膜传感器,并从β2稳定性定理出发推导出适用于薄膜传感器的微扰公式.该模型不仅能够清晰反映薄膜参量与包层模传播常量变化量之间的关系,而且在计算量和计算难度远低于四层波导模型情况下获得与严格求解结果相当的计算准确度.考虑数值计算本身引入的计算误差,该模型能够满足定性和半定量理论分析需要.最后通过长周期光纤光栅液态水膜的挥发实验对该模型进行了初步验证.     

Corrections to scaling for period doubling

We have studied a multiple scaling which describes corrections to scaling. For the period doubling in one-dimensional dissipative maps, two-dimensional areapreserving maps, and four-dimensional symplectic maps, the multiple scaling is seen to be well-obeyed, and new scaling factors have been found. The multiple scaling is also seen to be a very powerful tool for searching for scaling behavior.     

Polynomial Homogeneous Maps and Their Periods

We study the set of periods of the homogeneous polynomial maps $f: \R^n \to \R^n$ and $f: \C^n \to \C^n$ of degree $m>1$. For these complex maps, we also describe the number of invariant straight lines through the origin by $f^k$ for $k=1,2,\ldots$ and the dynamics of $f^k$ over them.     

沥青水浆不同结构分散剂的成浆性能研究

对脱油沥青水浆中不同结构分散剂的成浆性能和分散剂用量进行了研究,并探讨了分散剂添加方法对沥青水浆的影响,提出了结构相似相容的沥青水浆的分散剂研制和选型方法以及分散剂最可几摩尔用量的概念。实验结果表明硬沥青水浆最佳分散剂应是自身改性的磺酸盐,符合沥青水浆HLB值要求。分散剂最可几用量的确定,有助于根据工业具体要求确定最佳用量,即流动性、稳定性都满足工业要求的分散剂最经济用量。所用分散剂溶于水,多段添加的湿式粉碎成浆可制取高浓度、低粘度、稳定性好的硬沥青水浆。     

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.     

Isochronous centers of Lienard type equations and applications

In this work we study Eq. (E) with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F). We also classify a 5-parameters family of reversible cubic systems with isochronous centers.     

Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

In 1985 Franz Rothe [J. Reine Angew Math. 355 (1985) 129–138] found, by means of the thermodynamical equilibrium theory, an asymptotic estimate of the period of solutions of ordinary differential equations originated by predator–prey Volterra–Lotka model. We extend some of the Rothe's ideas to more general systems: