Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.     

Directional region control of the thermal fractal diffusion of a space body

We present a directional region control(DRC) model of thermal diffusion fractal growth with active heat diffusion in three-dimensional space. This model can be applied to predict the space body heat fractal growth and study its directional region control. When the nonlinear interference term and the inner heat source term are generalized functions, the relationship between the particle aggregation probability and the interference terms can be obtained using the norm theory. We can then predict the aggregation form of particles in different regions. When the nonlinear interference terms in the model are expressed as a trigonometric function and its composite function, our simulations show that the DRC method of thermal fractal diffusion is effective and has reference value for the directional control of actual fractal growth systems.     

A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations

It is known that weak interactions of two solitary waves in generalized nonlinear Schrödinger (NLS) equations exhibit fractal dependence on initial conditions, and the dynamics of these interactions is governed by a universal two-degree-of-freedom ODE system [Y. Zhu J. Yang, Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E 75 (2007) 036605]. In this paper, this ODE system is analyzed comprehensively. Using asymptotic methods along separatrix orbits, a simple second-order map is derived. This map does not have any free parameters after variable rescalings, and thus is universal for all weak interactions of solitary waves in generalized NLS equations. Comparison between this map’s predictions and direct simulations of the ODE system shows that the map can capture the fractal-scattering phenomenon of the ODE system very well both qualitatively and quantitatively.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Leading pollicott-ruelle resonances and transport in area-preserving maps

The leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wave number dependence determines the normal transport coefficients. In particular, a general exact formula for the diffusion coefficient D is derived without any high stochasticity approximation, and a new effect emerges: The angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The behavior of D is examined for three particular cases: (i) the standard map, (ii) a sawtooth map, and (iii) a Harper map as an example of a map with a nonlinear rotation number. Numerical simulations support this formula.     

Short-lived two-soliton bound states in weakly perturbed nonlinear Schrodinger equation

Resonant soliton collisions in the weakly discrete nonlinear Schrodinger equation are studied numerically. The fractal nature of the soliton scattering, described in our previous works, is investigated in detail. We demonstrate that the fractal scattering pattern is related to the existence of the short-lived two-soliton bound states. The bound state can be regarded as a two-soliton quasiparticle of a new type, different from the breather. We establish that the probability P of a bound state with the lifetime L follows the law P approximately L(-3). In the frame of a simple two-particle model, we derive the nonlinear map, which generates the fractal pattern similar to that observed in the numerical study of soliton collisions. (c) 2002 American Institute of Physics.     

Percolation on infinitely ramified fractals

We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.     

Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain

A single-channel color image encryption is proposed based on a phase retrieve algorithm and a two-coupled logistic map. Firstly, a gray scale image is constituted with three channels of the color image, and then permuted by a sequence of chaotic pairs generated by the two-coupled logistic map. Secondly, the permutation image is decomposed into three new components, where each component is encoded into a phase-only function in the fractional Fourier domain with a phase retrieve algorithm that is proposed based on the iterative fractional Fourier transform. Finally, an interim image is formed by the combination of these phase-only functions and encrypted into the final gray scale ciphertext with stationary white noise distribution by using chaotic diffusion, which has camouflage property to some extent. In the process of encryption and decryption, chaotic permutation and diffusion makes the resultant image nonlinear and disorder both in spatial domain and frequency domain, and the proposed phase iterative algorithm has faster convergent speed. Additionally, the encryption scheme enlarges the key space of the cryptosystem. Simulation results and security analysis verify the feasibility and effectiveness of this method.     

Aging in subdiffusion generated by a deterministic dynamical system

We investigate aging behavior in a simple dynamical system: a nonlinear map which generates subdiffusion deterministically. Asymptotic behaviors of the diffusion process are described using aging continuous time random walks. We show how these processes are described by an aging diffusion equation which is of fractional order. Our work demonstrates that aging behavior can be found in deterministic low dimensional dynamical systems.     

非线性Kronig-Penney超晶格的二维实映射分析

通过二维实映射研究非线性Kronig-Penney超晶格中的波输运特性，数值计算得到波矢一定情况下不同非线性系数的映射图、以及非线性系数一定情况下不同波矢的映射图.非线性Kronig-Penney超晶格中的非线性，对超晶格中波函数的Bloch波矢有明显的调节作用.随着非线性系数的增大，映射图由周期函数的有限个分立点变为准周期函数的一条闭合轨道、以及有规则或无规则的点分布. 关键词： 非线性Kronig-Penney超晶格 非线性效应 二维实映射 波矢     

Nonlinear dynamics in wurtzite InN diodes under terahertz radiation

We carry out a theoretical study of nonlinear dynamics in terahertz-driven n+ nn+ wurtzite InN diodes by using time-dependent drift diffusion equations.A cooperative nonlinear oscillatory mode appears due to the negative differential mobility effect,which is the unique feature of wurtzite InN aroused by its strong nonparabolicity of the Γ 1 valley.The appearance of different nonlinear oscillatory modes,including periodic and chaotic states,is attributed to the competition between the self-sustained oscillation and the external driving oscillation.The transitions between the periodic and chaotic states are carefully investigated using chaos-detecting methods,such as the bifurcation diagram,the Fourier spectrum and the first return map.The resulting bifurcation diagram displays an interesting and complex transition picture with the driving amplitude as the control parameter.     

分形多孔介质的粒子扩散特点(Ⅰ)

本文讨论了粒子在分形结构中的扩散,随机Sierpinski地毯上的扩散模拟表明:分形结构中的扩散已不再满足 Fick扩散定律。扩散速度较欧氏空间减慢了,即“扩散慢化”效应。而且可以看到由于结构的自相似特征,扩散过程也表 现出某种自相似性(标度性),分形结构和欧氏空间中的扩散特点是有本质的差异的。     

Anomalous diffusion in two-dimensional potentials with hexagonal symmetry

The diffusion process in a Hamiltonian dynamical system describing the motion of a particle in a two-dimensional (2D) potential with hexagonal symmetry is studied. It is shown that, depending on the energy of the particle, various transport processes can exist: normal (Brownian) diffusion, anomalous diffusion, and ballistic transport. The relationship between these transport processes and the underlying structure of the phase space of the Hamiltonian dynamical system is investigated. The anomalous transport is studied in detail in two particular cases: in the first case, inside the chaotic sea there exist self-similar structures with fractal properties while in the second case the transport takes place in the presence of multilayered structures. It is demonstrated that structures of the second type can lead to a physical situation in which the transport becomes ballistic. Also, it is shown that for all cases in which the diffusive transport is anomalous the trajectories of the diffusing particles contain long segments of regular motion, the length of these segments being described by Levy probability density functions. Finally, the numerical values of the parameters which describe the diffusion processes are compared with those predicted by existing theoretical models. (c) 2000 American Institute of Physics.     

Dynamics of conformal maps for a class of non-Laplacian growth phenomena

Time-dependent conformal maps are used to model a class of growth phenomena limited by coupled non-Laplacian transport processes, such as nonlinear diffusion, advection, and electromigration. Both continuous and stochastic dynamics are described by generalizing conformal-mapping techniques for viscous fingering and diffusion-limited aggregation, respectively. The theory is applied to simulations of advection-diffusion-limited aggregation in a background potential flow. A universal crossover in morphology is observed from diffusion-limited to advection-limited fractal patterns with an associated crossover in the growth rate, controlled by a time-dependent effective Péclet number. Remarkably, the fractal dimension is not affected by advection, in spite of dramatic increases in anisotropy and growth rate, due to the persistence of diffusion limitation at small scales.     

(G′/G)展开法与高维非线性物理方程的新分形结构

将(G′/G)展开法扩展到研究高维非线性物理方程的非行波解和分形结构.以(2+1)维变系数色散长波系统为例,构造出该系统的非行波解,对解中的任意函数进行适当的设置,发现了一类新的分形结构,即十字形分形结构.     

Flux front penetration in disordered superconductors

We investigate flux front penetration in a disordered type-II superconductor by molecular dynamics simulations of interacting vortices and find scaling laws for the front position and the density profile. The scaling can be understood by performing a coarse graining of the system and writing a disordered nonlinear diffusion equation. Integrating numerically the equation, we observe a crossover from flat to fractal front penetration as the system parameters are varied. The value of the fractal dimension indicates that the invasion process is described by gradient percolation.     

Supertracks,supertrack functions and chaos in the quadratic map

Chaotic behavior for the quadratic map is conjectured to be a bounding phenomenon based on state-space trajectories starting from the maximum in the map. These trajectories, called supertracks, have no sensitivity to initial conditions. The chaotic regime is found to be characterized recursively by these trajectories as functions of the map parameter.     

弱的参数周期扰动对一非线性系统安全域的影响与分形侵蚀安全域的控制

讨论了弱参数周期扰动对于非线性系统安全域的影响,在一定频率下的参数周期扰动将加速安全域的分形侵蚀,而在另一些频率的扰动下,将抑制安全域的分形侵蚀,并且存在着增进安全域的最优频率.提出了用弱参数周期扰动控制受到分形侵蚀的安全域的方法,并用Melnikov方法进行了分析.最后讨论了这种控制安全域的方法在实际环境中当具有外加噪声时的鲁棒性. 关键词： 分形吸引域边界 参数的周期扰动 控制     

调制无线电信号的分形特征研究

调制方式的不同以及信号受到功率放大器的非线性变换决定了调制无线电信号的形状存在着差异,提取这些差异特征是信号识别的基础. 基于分形理论提取信号特征是一种新的方法,本文模拟了在不同信噪比条件下,通过计算调制无线电信号的方差分形维以及Mandelbrot奇异分形维谱,将信号的调制特征以及非线性变换特征投射到分形特征空间. 数值仿真结果表明,不同调制方式、非线性变换对应了分形特征空间中的不同位置. 这些结果表明分形理论能够有效提取无线电信号的调制特征以及功率放大器的特征,对信号的调制方式以及个体特征识别具有一定 关键词： 无线电 信号 分形特征 识别     

Synthesis of fractal light pulses by quasi-direct space-to-time pulse shaping

We demonstrated a simple diffractive method to map the self-similar structure shown in squared radial coordinate of any set of circularly symmetric fractal plates into self-similar light pulses in the corresponding temporal domain. The space-to-time mapping of the plates was carried out by means of a kinoform diffractive lens under femtosecond illumination. The spatio-temporal characteristics of the fractal pulses obtained in this way were measured by means of a spectral interferometry technique assisted by a fiber optics coupler (STARFISH). Our proposal allows synthesizing suited sequences of focused fractal femtosecond pulses potentially useful for several current applications, such as femtosecond material processing, atomic, and molecular control of chemical processes or generation of nonlinear effects.