The Universal Plateau Structure of Lyapunov Exponents for Period Doubling Cascade Attractors

Using algebraic. analysis method for periodic orbits of Hknon map, we derive the boundary equations of stable window and Lyapunov exponent plateau region on the space of nonintegrability parameter A and dissipation parameter J. Ekom the real root of these equations, we obtain the plateau width of Lyapunov exponent W_p = A_p,max - A_p,min and the stable tvindorv width W_s = A_p,max - A_p,min for high periodic orbits. The calculated result of plateau structure ratio α4 = W_p/W_S for period-4 orbit agrees with the conjectural analytical formula: α4 = 2J²/(1+J⁴). Hence our result presents further evidence of universal dependence of Lyapunov exponent plateau structure on the dissipation parameter for period doubling cascade attractors of nonlinear system in transition from order to chaos.     

THE UNIVERSAL TRANSITION OF PLATEAU STRUCTURE OF LYAPUNOV EXPONENTS

The universal transition of Lyapunov exponents between conservative limit and dissipa-tire limit of nonlinear dynamical system is studied. It is discovered numerically and proved analytically that for homogeneous dissipative two-dimensional maps, along the equal dissi-pation line in parameter space, the Lyapunov exponents of attractor orbits possess a plateau structure and strict symmetry about its plateau value, The ratios between the plateau width and the stable window width of period 1-4 orbits for Henon map are calculated. The result shows that the plateau structure of Lyapunov exponents remains invariant for the attractor orbits belonging to a period doubling bifurcation sequence. This fact reveals a new universal transition behavior between order and chaos when the dissipation of the dynamical system is weakened to zero.     

一个新的自由能函数对介电高弹薄膜的多组等双轴预拉伸下力电耦合实验的预测

当介电高弹聚合物薄膜被施以面内等双轴预拉伸后, 受到厚度方向的电压作用时, 薄膜在力场和电场共同作用下产生大变形. 电场采用Maxwell应力分析, 力场采用橡胶弹性模型分析. 拟合这类变形的常用橡胶弹性模型主要有Neo-Hookean, Arruda-Boyce, Gent等模型. 这些模型对实验数据的定量拟合存在不同程度的偏差. 通过对实验数据的分析, 结合数学方法, 提出了一个新的自由能函数模型. 通过该模型对VHB4905介电高弹聚合物薄膜的多组等双轴预拉伸电力耦合实验进行拟合, 并以Neo-Hookean, Gent模型作为对照, 结果与实验数据拟合很好, 比对照模型的偏差明显缩小.     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).     

Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems

Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N×N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λ_n(L, t)=L^γ/ν f_n(tL^1/ν), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ₁, λ₂ and λ₆, we get the finite-size scaling form of the second moment correlation length ξ(L, t)=Lξ(tL^1/ν). It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.     

Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

We compute the Lyapunov exponent, the generalized Lyapunov exponents, and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, are used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. The Lyapunov exponent for the corresponding map agrees with the conjectured limit _map = -2 log(R) + C + O(R) and we derive an approximate value for the constantC in good agreement with numerical simulations. We also find a diverging diffusion constantD(t)logt and a phase transition for the generalized Lyapunov exponents.     

AlH~+离子5个Λ-S态和10个?态的光谱性质以及激光冷却的理论研究

利用考虑Davidson修正的内收缩多参考组态相互作用(icMRCI+Q)方法结合相关一致基组aug-ccpV6Z计算了AlH~+离子前两个离解极限对应的5个Λ-S态和10个?态的势能曲线.为了提高势能曲线的可靠性和精确性,计算中考虑了自旋轨道耦合效应、芯电子价电子相关和标量相对论修正以及将势能外推至完全基组极限.基于得到的势能曲线,获得了束缚和准束缚的4个Λ-S态和8个?态的光谱常数和振动能级,与已有的实验结果符合.计算了2(1/2)→X~2Σ_(1/2)~+和A~2Π_(3/2)→X~2Σ_(1/2)~+跃迁的跃迁偶极距.利用计算的精确的势能曲线和跃迁偶极矩,获得了2(1/2)~(第一势阱)(υ′=0, 1)→X~2Σ_(1/2)~+(υ′′)和A~2Π_(3/2)(υ′=0, 1)→X~2Σ_(1/2)~+(υ′′)跃迁的高度对角化分布的Franck-Condon因子(f_(00)和f_(11))和大的振动分支比;预测了2(1/2)~(第一势阱)(υ′=0, 1)和A~2Π_(3/2)(υ′=0, 1)态短的自发辐射寿命和窄的辐射宽度,这适合于AlH~+离子的快速激光致冷.所需的3束激光冷却波长都在紫外区域.这些结果表明了激光冷却AlH~+离子的可行性.此外,评估了自旋轨道耦合效应对光谱常数、振动能级和激光冷却AlH~+离子的影响.     

Modelling Accelerating Growth with IntermittentProcesses in Evolving Networks

In this paper, the acceleratingly growing network model with intermittent processes is proposed. In the growing network, there exist both accelerating and intermittent processes. The network is grown from the number of nodes m₀ and the number of links added with each new node is a nonlinearly increasing function m+aN^β(t)f(t), where N(t) is the number of nodes present at time t. f(t) is the periodic and bistable function with period T, whose values are 1 and 0 indicating accelerating and intermittent processes, respectively. Here we denote the ratio τ of acceleration time to whole one. We study the degree distribution p(k) of the model, focusing on the dependence of p(k) on the network parameters τ, T, m, a, N, and β. It is found that there exists a phase transition point k_c such that if k<k_c, then p(k) obeys a power-law distribution with exponent -γ₁, while if k>k_c, then p(k) exhibits a power-law distribution with exponent -γ₂. Moreover, the exponents γ₁ and γ₂ are independent of τ, T, m, a, and N, while they depend only on the parameter β. More interesting, the phase transition point is described by k_c=aN^β, which is equal to the value at which p(k) is maximum in GM model.     

The Criterion of Chaotic Pulsation in pulsating Stars

Up to now, the period-doubling bifurcation in variable star is identified through the existence of subharmonics in the poGer spectrum of light curves. Therefore the question is arisen: How to find a general criterion of period-doubling bifurcation in pulsating stars from theoretical analysis. It is discussed in this paper. We found that for the stellar pulsating mapping the bifurcation from a fixed point to a two-cycle occurs at parameter λ₁ = -0.63, i.e., adiabatic exponent Г₁ = 1.331.     

A Compact Exponential Operator for Generating Rotating and Squeezing Transformations

We point out that the compact exponential operator V≡exp[-i(λ₁Q²-λ₂P²)/2] (where λ₁, λ₂ are real;P and Q are momentum and coordinate operators, respectively;the units of h=m=ω=l is taken, m and ω are harmonic oscillator's mass and'frequency, respectively) is a generalized single-mode squeeze operator which can generate mixed squeezing and rotating transformations. The decomposition of V as a product of a squeeze operator and a rotating operator is derived through the technique of integration within an ordered product.The fluctuation in quadrature phases for the squeezed vacuum state generated by V is analyzed.     

Anti-control of chaos in Bose-Einstein condensates

The spatial structure of a Bose-Einstein Condensate (BEC) loaded into an optical lattice potential is investigated. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states into chaotic states. The maximal Lyapunov exponent is calculated as a function of modulation intensity and modulation frequency respectively, and the chaotic orbits associated with the positive Lyapunov exponents.      

A Possible Generalization of Jaynes-Cummings Model with Degenerate Spectrum

We propose a generalized Jaynes-Cummings model (JCM) for three two-level atoms and three-mode radiation field with the interaction being H₁=λ(a₁⁺a₂⁺σ_1-σ_2-+a₂⁺a₃⁺σ_2-σ_3-+a₁⁺a₃⁺σ_1-σ_3-+H.C.). This model is characteristic of possessing partly degenerate spectrum. Some transition process of this system is calculated.     

The Universality of the Critical Dynamics of the Kinetic Ising Model on the Regular Fractals

The form of the universal scaling law of the critical dynamic exponent, z = D_ƒ + 2/υ, is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents υ₀ = ∞ and υ = 1. So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, z = D_ƒ + R_max/2υ, on the site models of the regular fractals with R_min = 2. Meanwhile, we discuss Daniel Kandal's correction to the formula of the,critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.     

Periodic orbits in coupled Henon maps: Lyapunov and multifractal analysis

A powerful algorithm is implemented in a 1-d lattice of Henon maps to extract orbits which are periodic both in space and time. The method automatically yields a suitable symbolic encoding of the dynamics. The arrangement of periodic orbits allows us to elucidate the spatially chaotic structure of the invariant measure. A new family of specific Lyapunov exponents is defined, which estimate the growth rate of spatially inhomogeneous perturbations. The specific exponents are shown to be related to the comoving Lyapunov exponents. Finally, the zeta-function formalism is implemented to analyze the scaling structure of the invariant measure both in space and time.     

Low-temperature AC conductivity of Bi2Sr2CaCu2O8+δ

We report measurements of anamolously large dissipative conductivities, σ₁, in Bi₂Sr₂CaCu₂O_8+δ at low temperatures. We have measured the complex conductivity of Bi₂Sr₂CaCu₂O_8+δ thin films at 100–600 GHz as a function of doping from the underdoped to the overdoped state. At low temperatures there exists a residual σ₁ which scales with the T=0 superfluid density as the doping is varied. This residual σ₁ is larger than the possible contribution to σ₁ from a thermal population of quasiparticles (QP) at the d-wave gap nodes.     

Analytical solution for capacitive RF sheath

A self-consistent solution for the dynamics of a high voltage, capacitive radio frequency (RF) sheath driven by a sinusoidal current source is obtained under the assumptions of time-independent, collisionless ion motion and inertialess electrons. Expressions are obtained for the time-average ion and electron densities, electric field and potential within the sheath. The nonlinear oscillation motion of the electron sheath boundary and the nonlinear oscillating sheath voltage are also obtained. The effective sheath capacitance and conductance are also determined. It was found that: (1) the ion-sheath thickness S _m is √50/27 larger than a Child's law sheath for the DC voltage and ion current density; (2) the sheath capacitance per unit area for the fundamental voltage harmonic is 2.452 ϵ₀ /S_m, where ϵ₀ is the free space permittivity; (3) the ratio of the DC to peak value of the oscillating voltage is 54/125; (4) the second and third voltage harmonics are, respectively, 12.3 and 4.2% of the fundamental; and (5) the conductance per unit area for stochastic heating by the oscillating sheath is 2.98 (λ_D/S_m)^2/3 (e ²n₀/mu_e), where n ₀ is the ion density, λ_D is the Debye length at the plasma-sheath edge, and u_e is the mean electron speed     

Ground States of a Generalized Frenkel-Kontorova Model

We studied the modulated structures and commensurate-incommensurate transitions in a generalized Ftenkel-Kontorova (FK) model where the chain is composed of three kinds of atoms. For the classical ground states, the phase diagram and the phonon spectrum show the multi-atomic effects. For the quantum ground states, the multi-atomic effects are reflected by the distribution of atoms in external potential, the phase diagram, the correspondences between the ground states and the orbits of the area-preserving maps, as well as the phonon spectrum and coherence length. For the most incommensurate structure, the critical point K_c at which the transition by breaking of analyticity occurs does not vary monotonically with A. For the case of μ_A= μ_B = 1 and μ_c = 2, with the increasing of A, the critical point K_c first decreases, then increases. The minimum value of K_c occurs at nearly h = 0.035. When K < K_c, the phonon gap Ω_G = 0 and the Lyapunov exponent γ = 0, the system is in the sliding state. When K > K_c, the phonon gap and the Lyapunov exponent show great difference from those of the standard FK model.     

Mn(SiO2)3(M=Fe,Co,Ni;n=1-3)团簇的几何结构、光电性质及磁性

基于密度泛函理论中的广义梯度近似系统研究M_n(SiO₂)₃(M=Fe,Co,Ni;n=1-3)团簇的几何结构、光电性质和磁学性质.结果表明:Fe,Co原子相对于Ni原子更易于在(SiO₂)₃团簇上聚集;通过分析团簇的分裂途径及其产物,发现稳定性较好的氧化硅是一种很好的用于负载过渡金属"岛膜"的载体材料;M_n(SiO₂)₃团簇的能隙恰好位于近红外光谱范围内.通过磁性分析发现,该复合团簇的磁矩主要局域在过渡金属原子周围,而且,Fe₂(SiO₂)₃和Co₃(SiO₂)₃具有相对较大的磁矩,这主要源于过渡金属原子的d轨道间相互耦合.能隙和磁性两方面性质进一步肯定了二氧化硅磁性复合材料在医学界被用作光动力靶向治疗的可观前景.     

Magnetic phase transition in (Ba, Sr)Fe12O19 single crystals

The critical exponents β, γ and δ of single crystals of hexagonal ferrites BaFe₁₂O₁₉ and SrFe₁₂O₁₉ were determined from modified Arrott plots, scaling plots, ln(J) versus ln(μ₀H) plots and the method of Kouvel and Fisher. The results resemble those of the strong anisotropic ferromagnet Nd₂Fe₁₄B. Crossover effects are discussed by estimating the ranges within which dipolar and anisotropic interactions dominate, and by analyzing the temperature-dependent exponents β(T) and γ(T). In addition, the λ-line was determined from measurements in the hard direction, and the critical exponent ω was fitted.     

利用窄刻槽金属光栅实现石墨烯双通道吸收增强

构建基底/窄刻槽金属光栅/覆盖层/石墨烯结构,利用金属光栅激发的表面等离子体激元共振和窄光栅刻槽支持的法布里-珀罗共振,在可见光波段实现单层石墨烯的双通道吸收增强,并结合简化模型估算出双吸收通道所在位置.在波长462和768 nm处,石墨烯的光吸收效率分别为35.6%和40.1%,相比石墨烯本征光吸收率的增强均超过15.5倍.进一步研究发现由于短波处吸收增强源于金属光栅的表面等离子体激元共振,其吸收特性受覆盖层厚度、刻槽深度和宽度变化的影响较小;而由于长波处吸收增强源于窄刻槽中的法布里-珀罗共振,因此呈现出良好的角度不敏感吸收特性.