流体诱发水平悬臂输液管的内共振和模态转换(Ⅰ)

运用牛顿法导出水平悬臂刚性输液管的非线性动力学数学模型．为了对该模型进行理论分析,通过对各个相关实际物理量的量级定性分析,给出了模型中各个物理参数的量级．在此基础上,应用多尺度法首先得到输液管自由振动模态的特征函数,利用悬臂管的边界条件给出了特征值满足的特征方程, 发现管内流体速度可以诱发第一阶模态和第二阶模态3种形式的内共振分别是3∶1、2∶1和1∶1内共振, 从理论上解释了流速诱发水平悬臂输液管系统内共振的机理．由于3∶1内共振所对应的流速最小,因此这种形式的内共振是最先出现的,也是最重要的．     

五阶双色双向海洋表面波理论

将经典的"纯波动的三阶单色单向Stokes波理论"提升至"可包含环境均匀流效应的有限水深五阶双色双向海洋表面波理论".亦即,在已有三阶双色双波理论的基础上得到了自由表面位移、速度势和非线性振幅色散关系的第四阶、第五阶显式表达式.从中,将居于核心地位的"双色双向波的第五阶非线性振幅色散关系"又推广到"无穷多波中任意两两不同频率不同振幅相互作用波的非线性振幅色散关系".针对双色双向短峰波的典型特性,以若干个图表详加示之.     

快变海底和自由表面流共振生成的弱非线性Stokes波

运用WKBJ型摄动逼近法,对环境流同沙纹海底共振产生的自由表面水波的非线性效应进行了研究。沙纹海底由缓变平均水深部分和快变海底部分叠加构成。根据对快变海底波长的不同选取,可以相应地激发环境流同非平整海底的同步共振、超谐波共振和次谐波共振,由此产生自由表面波运动。对次谐波共振进行了详细考察。对于定常流自治动力系统,对可能出现的非线性各种稳态及其稳定性进行了探讨。假如环境流具有一个小振动分量,动力系统成为非自治的,则将发生混沌现象。     

一类分数阶非线性波方程的精确解及应用

基于分离变量的思想构造了分数阶非线性波方程含常系数的解的形式.在用待定系数法求解时,根据原方程确定假设解中的待定参数,得到具体解的表达式.利用该方法求解了3个非线性波方程,即分数阶CH(Camassa-Holm)方程、时间分数阶空间五阶Kdv-like方程、分数阶广义Ostrovsky方程.比较简便地得到了这些方程的精确解.文献中关于整数阶非线性波方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.对能够通过待定系数法求出精确解的分数阶微分方程所应满足的条件进行了阐述.     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

分数阶广义扰动热波方程的与泛函映射解

研究了一类分数阶广义非线性扰动热波方程.首先在典型分数阶热波方程情形下得到解,接着用泛函分析映射方法,求出了分数阶广义非线性扰动热波方程初始边值问题的任意次近似解析解.最后简述了它的物理意义.求得的近似解析解,弥补了单纯用数值方法得到的模拟解的不足.     

应用分数阶导数模拟桩屏障对粘弹性SH波的隔离

从粘弹性体波的三维分数阶本构方程出发,分析了软粘土中粘弹性P波和S波的频散效应,对比研究了非连续刚性桩和弹性桩对粘弹性SH波的隔离效果.利用有限差分法,分别模拟了不同桩间距与桩直径比、不同分数阶阶数及入射频率下振幅衰减系数的变化规律,并对比分析了刚性桩和弹性桩的隔震效果.数值实验表明,桩间距与桩直径比值越小,分数阶阶数越大,刚性桩隔震效果越好.对于某些特定的隔震区域,分数阶阶数越小,弹性桩隔震效果越好.     

有限深度两层流体系统中运动点源生成的内波及其与自由面的相互作用

基于水波动力学势流理论,以二流体系统中点源运动为基本模型,采用Green函数法,研究了分层流体中内部扰动诱生的内波对自由面的影响,探讨对合成孔径雷达成像起重要作用的自由面辐散场的变化规律。研究表明:当上下层流体密度跃变较大、源接近密跃层且Froude数Fr接近内波模式的临界值Fr₂时,内波模式导致的自由面辐散场强度与表面波模式的贡献相当,两者具有强烈的耦合作用,理论分析与实验结果定性一致。     

分数阶及整数阶Klein-Gordon方程的孤立波解研究

Klein-Gordon方程是量子力学领域的一类重要方程,它是薛定谔方程的一种相对论形式,包括分数阶和整数阶方程,寻求它的解有着重要的意义.利用一种较为实用的1/G展开法,对一类分数阶Klein-Gordon方程和相应的整数阶Klein-Gordon方程进行了求解,得到了丰富的行波解,包括孤立波解和扭曲波解,同时有代表性地选择一些解,来画出它们的图形并进行相图分析.另外,对所得到的整数阶与分数阶方程的解进行了对比,发现了它们的异同点.     

一类一维阵列的孤波特征

该文研究了一类由二自由度可积哈密顿系统构成的一维阵列的行波解,发现在长波极限下,问题可约化为分析哈密顿系统在扰动下的同异宿轨道的情形．当无扰系统具有共振时,利用能量──相方法,得到该系统存在同、异宿到不动点和周期轨的充分条件,在该条件下相应地一维阵列存在一组具有孤波特征的行波,同时给出了一个Ｎ脉冲孤立子波的例子．     

Primary resonance of Duffing oscillator with fractional-order derivative

In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude-frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude-frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude-frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.     

Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system

We develop an analytical technique to investigate the phenomenon of vibrational resonance in a fractional-order Duffing system with linear time delay feedback and driven by both low frequency and high frequency periodic signals. At first, the theoretical predication of the response amplitude at the low-frequency is obtained by the method of direct separation of slow and fast motions. Then, the bifurcation analysis is carried out based on three kinds of resonance behaviors. Further, influences of the high frequency signal, the fractional-order damping and the delay parameter on the vibrational resonance are discussed by both theoretical and numerical simulations. If the value of the fractional-order is a controllable parameter, the monotonicity of the response amplitude versus the value of the fractional-order depends on the amplitude of the high-frequency signal. If the delay parameter is a controllable parameter, the response amplitude with respect to the delay parameter presents periodic or quasi-periodic pattern, and it is similar to that in the integer-order differential system with linear time delay feedback. The good agreement between the analytical and numerical results indicates the validity of the theoretical predications.     

Nonlinear dynamics and chaos in a fractional-order financial system

This study examines the two most attractive characteristics, memory and chaos, in simulations of financial systems. A fractional-order financial system is proposed in this study. It is a generalization of a dynamic financial model recently reported in the literature. The fractional-order financial system displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in fractional-order financial systems with orders less than 3. In this study, the lowest order at which this system yielded chaos was 2.35. Period doubling and intermittency routes to chaos in the fractional-order financial system were found.     

Abundant bursting patterns of a fractional-order Morris–Lecar neuron model

In this paper, a fractional-order Morris–Lecar (M–L) neuron model with fast-slow variables is firstly proposed. The fractional-order M–L model is a generalization of the integer-order M–L model with fast-slow variables, where the fractional-order derivative is used to characterize the memory effect and power law of membranes. Then the bursting patterns of the new model are investigated by using the bifurcation theory of fast-slow dynamical systems. Numerical simulation shows that the new model exhibits some bursting patterns that appear in some common neuron models with properly chosen parameters but do not exist in the corresponding integer-order M–L model. Further, on the basis of a comparison of the nonlinear dynamics between the fractional-order M–L model and the integer-order M–L model, we show that the fractional-order derivative can activate the slow potassium ion channel faster and play an important role to modulate the firing activity of the new model.     

Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller

This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.     

Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models

In this paper we are concerned with the fractional-order predator-prey model and the fractional-order rabies model. Existence and uniqueness of solutions are proved. The stability of equilibrium points are studied. Numerical solutions of these models are given. An example is given where the equilibrium point is a centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.     

Van der pol情绪混沌模型的滑模同步

研究了整数阶分数阶van der pol情绪混沌模型的滑模同步问题,利用分数阶微积分给出了情绪模型的主从系统取得同步的充分条件,研究表明,一定条件下,Van der pol情绪模型的主从系统能够达到同步,数值仿真验证了该方法的可行性.     

基于双分数阶变分光流模型的运动目标检测方法

变分光流法是常用的运动目标检测方法,应用场景中的光照变化会极大影响现存变分光流法的稳定性及准确率,提出基于分数阶的变分光流模型来提高光照变化鲁棒性.该模型将分数阶导数同时应用于经典变分光流模型的数据项及平滑项中;根据图像的数据特征,采用图像函数及光流向量函数的有限离散二重分数阶导数近似拟合求导结果计算光流向量,进而将光...     

Stability of non-constant steady-state solutions for bipolar non-isentropic Euler–Maxwell equations with damping terms

In this article, we consider the periodic problem for bipolar non-isentropic Euler–Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler–Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler–Maxwell/Poisson equations.     

空间分数阶电报方程的格子Boltzmann方法

应用格子Boltzmann方法（LBM）对Riemann Liouville空间分数阶电报方程进行了数值模拟研究.首先,将分数阶算子中的积分项进行离散化处理,并进行了收敛阶分析.然后,构建了带修正函数项的一维三速度(D1Q3)的LBM演化模型.利用Chapman Enskog多尺度技术和Taylor展开技术,推导出各平衡态分布函数和修正函数的具体表达式,准确地从所建的演化模型恢复出宏观方程.最后,数值计算结果表明该模型是稳定、有效的.