一维单峰映象的拓扑熵

在对Logistic映象数值计算的基础上,我们分析了一维单峰映象的逆轨道结构,证明了不同参数处逆轨道总数N(n)随求逆次数n而变化的递推公式。借此解析地求得了在倍周期区中h(f)≡0;在U序列RLR²¹的m=3+2l周期点上h(f)=logα_mp,其中α_mp为方程α^m-2α^m-2-1=0的最大实根;在2^j-1常和2j带交界处h_j(f)=(1/2)^jlog2,由此可得聚点μ_∞处拓扑熵的标度指数t=0.449806…。在此基础上,我们还求得了混沌区的周期窗口,U序列RL^aR^b所对应的各点处的拓扑熵,以及h_R*Q(f)=(1/2)h_Q(f)的关系。证明是在M.S.S.规则和“*”乘法则的基础上进行的。所以本文的结果对一维单峰映象是普适的。 关键词：     

一类分段光滑映象中有新特征的激变

讨论了描述一类电子张弛振子的分段光滑映象中的两种不连续性导致激变的特性.一种激变发生的机理是一个混沌吸引子吸引域内的不稳定周期轨道与映象的不连续区碰撞;而另一种激变的机理是一个混沌吸引子与两个映象的不连续区构成的“映孔”碰撞.发现第一种激变的平均层流相长度的标度律为〈τ〉∝^-1.8,层流相长度分布的标度律为P(τ)=1〈τ〉·exp-τ〈τ〉,而第二种激变的标度律分别为〈τ〉∝exp(k^-1/2)和P(τ)=1〈τ〉exp-τ〈τ〉. 关键词：     

Belousov-Zhabotinsky反应体系中的波与稳定性

我们在Oregonator模型的基础上,取速率常数k₆、理想配比系数f及波数k为参变量,首先讨论了B-Z反应体系的稳定性,并得到了下列各量的表达式:空间周期结构的临界速率常数k₆^c、短波临界波数k_s^c与长波临界波数k_L^c,时空周期结构的临界速率常数k₆^c及临界波数k_c等。而后,我们给出了B-Z反应体系的临界频率λ_c,讨论了体系在临界状态附近的行为,并得到了触发波速度u_T及相位波速度u_P的表达式。本文的结论是,在一定条件下,不搅拌的B-Z反应体系能在临界状态附近呈现稳定的波群或者波包,触发波是均匀体系中的波包,相位波是有微小梯度的非均匀体系中的波包。u_T=2ηDk_c,D为HBrO₂的扩散系数,η为[H⁺]的缓变函数(在实验观测范围内,η≈0.1);u_P=v,v为相速度。 关键词：     

研究强迫非线性振子中倍周期分岔和“混乱”现象的分频采样方法

采用分频采样方法对于在周期外力作用下的非线性振子进行数值研究,达到可与离散映象相比拟的高分辨力。首次为常微分方程组描述的系统确定了高达8192分频的倍周期分岔序列和相应的“混乱”带的序列,并证实存在着嵌在混乱带中的二阶和三阶分岔序列。讨论了分频采样方法的优点和局限性,以及使用这一方法时应注意的问题。 关键词：     

一维准周期体系中的电子延展态

本文采用动力学映象方法,研究了作为泛Fibonacci序列一个子集的一系列一维准周期体系的电子性质,该一维准周期体系S_∞由递推公式S_l+1=S_l^2j-1S_l-1²ⁱ构造,其中l≥1,i,j为正整数,初始条件为S₀和S₁,结果表明,该一维准周期体系无论是对角模型和非对角模型,都存在电子延展态。 关键词：     

Josephson结的动力学行为(Ⅰ)

本文对射频电流驱动下的包含干涉项电流的Josephson结方程进行了大量的数值研究工作。我们发现,对应于不同的参数范围,分别出现混沌行为、倍周期分岔序列、混沌带的反序列以及阵发混沌现象。我们也计算了2ⁿP序列的收敛因子δ_n和功率谱中2^k与2^k+1分频的平均峰高之比Φ(k)/Φ(k+1)。另外,还研究了周期解的对称性以及它与通向混沌途径的关系。 关键词：     

不可微不可逆映象中的V型阵发

一个既不可微(或不连续)又不可逆的一维映象可以展示一种新型的阵发。它的机制是稳定不动点与映象不可微或不连续点碰撞而消失。这种阵发可以在该不动点附近的线性化映象本征值绝对值在阵发前为小于1的任何值的情况发生,因而可能突然出现在倍周期分岔序列中途任一部分,使序列中断进入混沌。在稳定不动点消失后映象产生的阵发时间序列中,层流相长度呈现与外控参数距临界值距离的对数依赖关系。这种新型标度规律不依赖于映象的细节。作者认为这种阵发应广泛存在于许多实际系统之中。 关键词：     

一个二维分段光滑映象中的边界激变

解析地讨论一个冲击振子模型中的边界激变特征,证明了这类分段光滑二维映象中激变的生 存时间标度律为τ-ε^-γ,而γ=ln|β₂|ln|β₁β2|(β₁,β₂分别是混沌吸引子吸引域边界上鞍点的不稳 定和稳定本征值),这与处处光滑二维映象中相应的规律完全不同. 关键词： 激变 分段光滑映象 生存时间标度律     

非线性环形光学腔中分岔混沌的新行为

利用数值模拟方法研究了非线性环形光学腔中的Ikeda方程,发现:长延时下,分岔图的拓扑结构、分岔方式、收敛速率以及周期窗口的结构都随参数B,φ₀的取值不同而异。短延时下,通过适当近似,将Ikeda方程化为带有一定记亿效应的迭代映象,所得到的近似系统显示出非倍周期分岔行为,并伴有一些奇特的周期窗口,混沌区中的无限嵌套的自相似结构变得模糊,不易分辨。 关键词：     

推广Bethe公式：Au50+离子的偶极激发的碰撞强度和速率系数

快速算法结合推广的Bethe公式,可以为应用研究提供便于使用的偶极激发碰撞强度和速率系数.用准相对论平面波Born(QRPB)近似计算高能区Au⁵⁰⁺离子n₀l₀→nl偶极激发的碰撞强度,给出Bethe公式中动量转移截断参数k₀,从而确定激发过程的高能行为.快速计算方法中采用了Cowan所发展的准相对论方法,用统一的Hartree-Fock-Slater势计算束缚和连续态电子波函数.用准相对论扭曲波(QRDW)近似计算阈值附近的碰撞强度并外推阈值处的碰撞强度Ω₀,然后拟合到高能碰撞强度上.对于特殊情况还需增加三倍阈值点a的计算,用三个参数Ω₀,Ω_a和k₀拟合出全能域(出射电子能量ε_b=0—∞)的碰撞强度.由此可以得到全部温度范围(电子温度T_e=0—∞)的速率系数<σv>.这样得到的Ω和<σv>,在相应的感兴趣的能量和温度范围内有合理的精度. 关键词：     

Period doubling bifurcations and universality in conservative systems

We have studied numerically a sequence of period doubling bifurcations of periodic orbits of Hénon's conservative two-dimensional mapping. In agreement with other recent studies, we also find evidence that such sequences possess universality properties, similar to the ones observed for dissipative systems. The corresponding universal constants, however, have significantly different values in each case. We suggest that a possible explanation for this discrepancy may be the fact that conservative systems, owing to their measure preserving property, do not become asymptotically one-dimensional, as dissipative systems do.     

Universal Scalings in Homoclinic Doubling Cascades

Cascades of period doubling bifurcations are found in one parameter families of differential equations in ℝ³. When varying a second parameter, the periodic orbits in the period doubling cascade can disappear in homoclinic bifurcations. In one of the possible scenarios one finds cascades of homoclinic doubling bifurcations. Relevant aspects of this scenario can be understood from a study of interval maps close to x↦p+r(1 −x ^β)², β∈ (?,1). We study a renormalization operator for such maps. For values of β close to ?, we prove the existence of a fixed point of the renormalization operator, whose linearization at the fixed point has two unstable eigenvalues. This is in marked contrast to renormalization theory for period doubling cascades, where one unstable eigenvalue appears. From the renormalization theory we derive consequences for universal scalings in the bifurcation diagrams in the parameter plane. Received: 16 June 1999 / Accepted: 24 April 2001     

Lyapunov number for a noisy 2 n cycle

A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2 ⁿ cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.     

有磁辫等离子体的双撕裂模研究

在电磁流体力学范围内研究了与磁辫相联系的反常电子粘滞性效应所驱动的双撕裂模的线性行为。采用分区处理方法得到了该模式的一个近似色散关系。如果在x=±x_s处的两个有理面间的距离足够小[x_s/α《(k_yα)^-11/15R^-11/15],则所得增长率的变化规律为τ_v^-1/5,随着该距离的增大,增长率过渡为τ_r^-1/3定标。τ_v^-1/5定标被证明为与撕裂层中不变φ近似的失效有关。 关键词：     

Statistical dynamics at critical bifurcations in Duffing-van der Pol oscillator

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance Σ_n(q) of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, Σ_n (q) associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of Σ_n(q) is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variablex is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent λ(X,L) and the local mean valuex(L) of the coordinatex calculated after everyL time steps are found to approach zero in the limitL → ∞ asL ^-Β. Β is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of ak-step difference quantity δx_k = x_i+k x _i.     

Superfluorescence and amplified spontaneous emission in an inhomogeneously broadened medium

The emission from a high gain pencil-shaped volume of inhomogeneously broadened (T¹₂) and initially completely inverted two-level atoms is described fully quantum-mechanically taking into account propagation along the pencil axis. The emission is shown to be superfluorescence for T¹₂ ? √τRτD and amplified spontaneous emission for τR ? T¹₂ ? √τRτD, where τR and τD are the collective decay time and the delay time of pure (T¹₂ = ∞) superfluorescence, respectively.     

Statistical properties of a dissipative kicked system: Critical exponents and scaling invariance

A new universal empirical function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal empirical function describes remarkably well a phase transition from limited to unlimited growth of the average action.