谐和激励与有界噪声作用下具有同宿和异宿轨道的Duffing振子的混沌运动

研究了具有同宿轨道、异宿轨道的双势阱Duffing振子在谐和激励与有界噪声摄动下的混沌运动.基于同宿分叉和异宿分叉，由Melnikov理论推导了系统出现混沌运动的必要条件及出现分形边界的充分条件.结果表明：当Wiener过程的强度参数大于某一临界值时，噪声增大了诱发混沌运动的有界噪声的临界幅值，相应地缩小了参数空间的混沌域，且产生混沌运动的临界幅值随着噪声强度的增大而增大.同时数值计算了最大Lyapunov指数，由最大Lyapunov指数为零从另一角度得到了系统出现混沌运动的有界噪声的临界幅值，发现在Wi 关键词： 混沌 同宿和异宿分叉 随机Melnikov方法 最大Lyapunov指数     

晶体摆动场辐射系统的全局分叉与混沌行为

引入正弦平方势,在经典力学框架内和偶极近似下,考虑到运动阻尼和非线性影响,把粒子在晶体摆动场中的运动方程化为具有阻尼项和受迫项的广义摆方程.利用Jacob椭圆函数和椭圆积分分析了无扰动系统的相平面特征,并解析地给出了系统的解和粒子振动周期; 进一步利用Melnikov方法分析相平面上三类轨道的分叉性质和进入Smale马蹄意义下的混沌行为,找到系统的全局分叉与系统进入混沌的临界条件.结果表明,系统的临界条件与它的物理参数有关,只需适当调节这些参数就可以原则上避免、抑制分叉或混沌的出现. 关键词： 晶体摆动场辐射 Melnikov方法 分叉 混沌     

沟道效应的运动阻尼与系统走向混沌的临界特征

利用正弦平方势,把粒子运动方程化为具有运动阻尼和周期调制的摆方程.利用Melnikov方法分析了系统的全局分叉和混沌行为，指出了沟道辐射本底增强和沟道效应的无规行为与混沌之间的相关性. 关键词： 沟道效应 摆方程 分叉 混沌     

晶体摆动场辐射系统的全局分叉与混沌行为

引入正弦平方势,在经典力学框架内和偶极近似下,考虑到运动阻尼和非线性影响,把粒子在晶体摆动场中的运动方程化为具有阻尼项和受迫项的广义摆方程.利用Jacob椭圆函数和椭圆积分分析了无扰动系统的相平面特征,并解析地给出了系统的解和粒子振动周期; 进一步利用Melnikov方法分析相平面上三类轨道的分叉性质和进入Smale马蹄意义下的混沌行为,找到系统的全局分叉与系统进入混沌的临界条件.结果表明,系统的临界条件与它的物理参数有关,只需适当调节这些参数就可以原则上避免、抑制分叉或混沌的出现.     

再入飞行器失稳的分叉理论分析与数值仿真验证

飞行器再入大气层时的姿态稳定性事关飞行安全, 是气动设计的关键问题之一.文章采用非线性自治动力系统分叉理论, 耦合求解非定常Navier-Stokes方程和俯仰运动方程, 研究了钝体和细长体两类航天飞行器再入过程单自由度俯仰运动失稳问题.研究表明, 航天飞行器再入时, 如果仅有1个配平攻角, 随Mach数降低, 其配平攻角处的俯仰姿态失稳一般对应于Hopf分叉, 并存在亚临界Hopf分叉和超临界Hopf分叉两种失稳形态; 如果再入时随着Mach数的降低, 其配平攻角由1个演化至多个(一般为3个), 其配平攻角处的俯仰姿态失稳形态将更为复杂, 可能发生鞍结点分叉形态的刚性失稳行为;随Mach数的进一步降低, 其俯仰运动还可能进一步发生Hopf分叉和同宿分叉.      

非各态历经噪声的产生及其应用

用n阶时间导数噪声作为内部热噪声驱动自由粒子运动时,若该噪声所对应的非各态历经强度b?=0,且系统初始处于热平衡,则此时系统的长时稳态速度可以作为非各态历经噪声使用.非各态历经噪声具有谱密度在零频处发散的特点,一维半无界耦合简谐振子链对与之相连的气体分子的作用具有非各态历经噪声的特点,是非各态历经噪声的实例.最后将非各态历经噪声作为外噪声驱动倾斜周期势中的粒子运动,系统出现扩散指数α2的超弹道扩散现象.同时发现系统的速度分布将呈现出迁移态和锁定态两种不同状态,并且处于迁移态的粒子的速度分布为双峰.迁移态中双峰的出现是一种比较新奇的现象.     

带电微粒在滤膜附近的定向运动

滤膜在过滤和分离领域具有非常广泛的使用,但滤膜对溶液中的组分具有吸引或排斥作用,导致粒子产生定向运动,进而在膜附近产生粒子富集或“粒子禁区”等截然不同的效应,目前对此认识尚不清楚,粒子浓度分布不能实时准确测量是其中最大的困难.本文以表面带负电荷的胶体粒子为模型物质,利用胶体粒子结晶后粒子浓度可以实时原位测量的优势,研究了纤维素滤膜对粒子运动行为的影响规律.研究结果发现带电微粒会自发地在滤膜表面产生富集,滤膜中微量阴阳离子释放导致的扩散泳效应是微粒朝向滤膜产生定向运动的主要原因,基于扩散泳和粒子扩散两种机制构建了粒子运动方程和模型,数值计算结果和实验结果定性一致.此外,粒子除了朝向滤膜的纵向运动之外,壁面电渗流和横向扩散还会使粒子朝向壁面产生迁移,导致管壁粒子数的增多.     

固体粒子在直流式除尘器中湍流脉动的数值模拟

本文对直流除尘器涡室内固体粒子的湍流脉动现象进行了数值分析,通过气体速度随机脉动谱,把气相湍流运动对固体粒子运动的影响引入粒子的运动平衡方程中,用拉格朗日法模拟了粒子的轨迹及其扩散运动,应用四阶龙科库塔方法求解粒子的运动方程。计算结果表明对粒子的数值模拟可以较好地预测除尘器的性能,如除尘器的切割粒径。     

活性布朗粒子运动的稳态解

拟可积Hamilton系统随机平均法可以用来研究活性布朗粒子运动.介绍了该随机平均法，利用它详细求解了布朗粒子运动的动力学方程，该方程描述了活性布朗粒子在平面上的运动，粒子受到的激励是Gaussian白噪声，受到的阻尼是Schienbein-Gruler速度依赖的磨擦模型.通过与数字模拟和与实验数据的比较，证明所得稳态解正确.对于Rayleigh和Erdamnn速度依赖的磨擦模型，也给出了稳态解. 关键词： 活性布朗粒子 拟可积Hamilton系统随机平均法 可积性 稳态解     

一类复杂流行病学模型的混沌研究

研究了一类周期变化的非线性复杂发病率的广义流行病学模型(SIR(susceptible, infected, recovered)模型). 通过一系列坐标变换将原模型转化为Hamilton系统,运用Melnikov方法证明了该系统存在混沌运动,给出了发生同宿分岔的条件,并用数值模拟验证了上述结果. 关键词： SIR(susceptible infected recovered)模型 混沌运动 Melnikov方法 同宿分岔     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Determination of the exact range of the value of the parameter corresponding to chaos based on the Silnikov criterion

Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.     

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.     

Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation

Systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable-mass, subjected to anti-phase support motion at primary resonance, is accomplished. Upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics, an improved experimental apparatus is used to make it technically accessible. Results obtained by varying three control parameters, namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system, allow us to characterize in-depth various occurring classes of motion in terms of time and spatial complexity, to describe peculiar and/or persistent features of transition to nonregular dynamics, and to trace them back to a canonical scenario from bifurcation theory. Variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results, and overall behaviour charts are built in the excitation parameter space. Considering the temperature as a controllable parameter shows to be fundamental for: (i) indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent, (ii) qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence-Hopf bifurcation normal form, and (iii) determining system response not only in the strict neighbourhood of the organizing divergence-Hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low-dimensional homoclinic chaos.     

沉积工艺对二氧化锆薄膜生长特性影响的研究

利用反应离子束溅射、反应磁控溅射和电子束蒸发在K9基底上沉积ZrO2薄膜，并用原子力显微镜对薄膜表面形貌进行测量。通过数值相关运算，对不同工艺条件下薄膜生长界面进行定量描述，得到了薄膜表面的粗糙度指数、横向相关长度、标准偏差粗糙度等参量。由于沉积条件的不同，薄膜生长具有不同的动力学过程。在反应离子束溅射和反应磁控溅射沉积薄膜过程中，薄膜生长动力学行为均可用Kuramoto-Sivashinsky方程来描述，电子束蒸发制备薄膜的过程可以用Mullins扩散模型来描述，并发现在沉积薄膜过程中基底温度和沉积过程的稳定性对薄膜表面特征影响很大。     

Exploiting the Hamiltonian structure of a neural field model

We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model’s behaviour. The lack of robustness of the model to small perturbations in parameters is surprising, and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the time-independent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves. By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.     

Basic structures of the Shilnikov homoclinic bifurcation scenario

We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits.     

参外联合激励复合非线性振子的分岔分析

讨论了参外联合激励复合非线性振子的动力学行为.对其定常解在一阶近似下的方程进行了局部分岔分析,给出了简单分岔和Hopf分岔发生的条件,并通过对近似方程和原系统的数值模拟加以验证.分析了多种参数对该振子动力学行为演化过程的影响.根据全局分岔理论探讨了该振子在不同条件下发生同宿、异宿分岔的必要条件,其结论与数值计算的结果大致符合. 关键词： 复合非线性振子 局部分岔 全局分岔 混沌     

有扩散不稳定性的四阶反应-扩散系统的空间周期结构

本文利用分叉理论研究一个含参数的有扩散不稳定性的四阶反应-扩散系统的空间周期结构问题,其中扩散项服从Cahn-Hillard广义扩散定律。首先通过稳定性和奇异性的分析得到从空间均匀定态产生空间周期定态的判别准则,然后利用奇异摄动方法获得这个空间周期定态的二阶近似解。本文的理论结果很好地符合文献[3]中的数值结果。 关键词：     

Slow passage through a transcritical bifurcation for Hamiltonian systems and the change in action due to a nonhyperbolic homoclinic orbit

One-degree of freedom conservative slowly varying Hamiltonian systems are analyzed in the case in which a saddle-center pair undergo a transcritical bifurcation. We analyze the case in which the method of averaging predicts the solution crosses the unperturbed homoclinic orbit at the precise time at which the transcritical bifurcation occurs. For the slow passage through the nonhyperbolic homoclinic orbit associated with a transcritical bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits surrounded by autonomous nonlinear saddle approaches. The change in action is computed by matching these solutions to those obtained by averaging, valid before and after crossing the nonhyperbolic homoclinic orbit. For initial conditions near the stable manifold of the nonhyperbolic saddle point, one saddle approach has particularly small energy and instead satisfies a nonautonomous nonlinear equation, which provides a transition between nonhyperbolic homoclinic orbits, centers, and saddles. (c) 2000 American Institute of Physics.