A short historical account of period doublings in the pre-renormalization era

I will shortly review the history of experimental and theoretical findings on period doubling until the discovery of the quantitative universal properties of the infinite period-doubling cascade.     

The center conditions and local bifurcation of critical periods for a Liénard system

In this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a Liénard system with forth damping. Calculating the singular point values and period constants, we find all center conditions and isochronous center conditions. Moreover, the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers.     

Period interaction on diffraction efficiency of blazed transmission gratings

The period interaction on diffraction of the blazed transmission gratings is analyzed with a modified extended scalar theory. For one certain period, the lights reflected from the neighboring structures can be refracted by the grating facets and form two extra fields on the bottom facet of blazed transmission gratings. The effects of this period interaction versus several diffraction orders for a fixed fabrication error of blazed transmission gratings with intermediate structures are discussed for both TE and TM polarizations. The results have been compared with those obtained with finite-difference time-domain method.     

One-dimensional photonic crystal design

In this article we present a method to determine the band spectrum, band gaps, and discrete energy levels, of a one-dimensional photonic crystal with localized impurities. For one-dimensional crystals with piecewise constant refractive indices we develop an algorithm to recover the refractive index distribution from the period map. Finally, we derive the relationship between the period map and the scattering matrix containing the information on the localized modes.     

基于TOA信息的脉冲星信号周期估计方法

针对脉冲星信号周期估计需逐步搜索、计算量大等问题,提出脉冲星信号周期直接估计算法.根据周期不确定误差对时域上脉冲到达时间(Pulse Time of Arrival,TOA)估计方法的影响,推导了基于TOA信息的脉冲星信号周期估计方法的数学模型.该方法将一组观测到的脉冲光子到达时间序列(Photon Time of Arrivals,TOAs)进行等时间间隔分段,在时域上对每段TOAs进行相位估计获得相应的TOA信息,并根据建立的TOA信息与周期误差的关系采用最小二乘原理估计脉冲星信号周期.对周期准确性的评价不同于传统的依赖折叠轮廓的好坏,而是以TOA与时间图像的斜率作为依据,这一依据更直观、易于理解.理论分析与物理、计算机仿真数据结果表明,本文所提的周期估计方法能够根据短时间脉冲光子到达时间序列获得高精度和高分辨率的脉冲信号周期,有利于X射线脉冲星导航的工程应用.     

PDMS梯度光栅结构制备技术研究

由于传统方法制作的梯度光栅,工艺条件苛刻,制作过程复杂,难以控制,制作成本高,周期较长,提出了一种成本低、工艺简单、可大量制备梯度光栅的工艺方法,采用基于刚性薄膜/柔性衬底的自组装工艺和氧等离子体(Plasma)的方法制备了微米尺度的梯度光栅,利用Plasma时间的可控性和聚二甲基硅氧烷(PDMS)优异的弹性制得所需要尺寸的光栅。首先在聚乙烯对苯二酸脂(PET)薄膜上旋涂一层PDMS薄膜,待PDMS薄膜固化后将双层薄膜弯曲并用Plasma处理,在其表面生成一层刚性氧化层,借助柔性的PET对刚性层施加均匀应力,当应力超过临界值时,在PDMS基底上自组装形成光栅褶皱结构。由于弯曲时预应力的变化,所以在PDMS薄膜上会形成周期和高度呈阶梯状的的光栅褶皱,也就是梯度光栅。采用可见光作为梯度光栅的性能测试光源,选用一级衍射光作为检测对象,从图谱中可以看出以PDMS为基底制备的光栅具有很好的衍射效应,并可实现很好的分光效果。实验表明：梯度光栅具有明显的衍射现象,并且衍射角变化显著,可广泛用于应力测量。这种方法制备的柔性梯度光栅也可以作为微型应变装置来检测应力的变化,未来有望用于微型光谱仪、扫描仪、光通讯等领域中。     

Period adding bifurcation in a logistic map with memory

We introduce a single step memory dependence in the fully chaotic logistic map. This makes it a two dimensional system in general. However, we show that by using composite functions to define two one dimensional maps, it is possible to obtain some analytic results for the bifurcation structure. Numerical results support the calculated bifurcation scheme and, in addition, yield a further insight which allows the calculation of the convergence ratio for a new period adding scenario.     

Imprimitive matrices of zeros and ones with extremal numbers of ones

In [X. Zhan, Extremal numbers of positive entries of imprimitive nonnegative matrices, Linear Algebra Appl. 424 (2007) 132–138], Zhan determined the maximum and minimum numbers of positive entries of irreducible nonnegative matrices. In this paper, we characterize the irreducible (0,1) matrices with the maximum and minimum numbers of positive entries.     

Two novel methods for vibration diagnosis to characterize non-linear response

Poincaré maps have been proved to be a valuable tool in the analysis of non-linear dynamical systems, which usually reduce a continuous phase flow into a two-dimensional discrete map. However, they may be inconvenient for reflecting some characteristics of the system response. In this paper, two novel methods, using the period sampling peak-to-peak value (PSP) diagram and the modified Poincaré map, are presented for characterizing different types of non-linear response. These two methods take advantage of some parameters of the response, such as the peak-to-peak value within an exterior excitation period and the mean value of the displacement. In the PSP diagram method, a two-dimensional graph is plotted by taking the peak-to-peak value as ordinate and the sequential periodically sampling number as abscissa. On the other hand, the modified Poincaré map takes the mean value of the velocity within an exterior excitation period as ordinate and the relevant mean value of the displacement as abscissa. The non-linear responses of a Duffing system, a pendulum with circular motion support and an oscillating circuit are studied by these methods. We also studied the intermittent chaos of the Lorenz system by the PSP diagram method. The PSP diagram is a set of mapping points, which form: a straight line for a one-period response; multi-straight lines for a multi-period response; orderly periodic curves for a quasi-period response; long lines interrupted by transitoriness confusion points for intermittent chaos; and totally out-of-order points for chaos. The figures for the modified Poincaré maps for the period, multi-period, quasi-period responses and chaos are almost identical to those for the Poincaré maps, but the modified maps take more sampling points and can reflect the mean values of the responses. Some numerical results are given based on these methods to show their efficiency in distinguishing different non-linear responses.     

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.