规范形网络中的混沌吸引子

讨论多余维Hopf分叉三阶规范形的普适开折形成的网络更进一步的复杂动力学行为．通过对余维二Hopf分叉的规范形网络多级分叉的分析，发现在参数空间的某个区域会出现二环面，将S形非线性加入规范形网络，在出现二环面的区域内可以出现混沌．本文给出了该混沌吸引子的相图及其二阶Poincare映射的图景．由这些图可以看到该混沌吸引子具有非常奇妙的形态：某些二阶Poincare映射像一只逼真的蝴蝶．     

神经网络动力系统吸引子对称性的变化

用余维２的双Ｈｏｐｆ分叉的规范形方程设计了期望存储振荡型记忆模式的模拟四阶关系神经网络，所设计的网络向量场具有中心对称性。研究表明该网络发生二阶双Ｈｏｐｆ分叉后可以出现不变二环面。观察二环面上的销相运动，发现了系统出现对称破裂的规律，邓把双Ｈｏｐｆ分叉的两个频率比表示成既约分数的形式，当该分数的分子和分母均为奇数时，网络的吸引子保持对称性，而当分子主分母中任一个为偶数时，就会发生对称破裂。     

神经网络的分叉理论设计方法

本文用分叉理论的规范形方程设计和综合期望贮存静、动态记忆模式的神经网络。对于期望贮存静态记忆模式的网络，该规范形方程为叉形分叉的；若期望贮存的记忆模式是周期振荡形式，该规范形方程为高余维数Ｈｏｐｆ分叉的，由满足设计约束的规范形系数得到的突触连接系数可以保证期望贮存的记忆模式都能成功地存贮于所设计的网络，且是网络仅有的吸引子，没有伪吸引子，吸引域的范围足够大。     

双同步混沌吸引子动力系统中的筛形域

解析证明耦合映射混沌同步系统中的两个同步混沌吸引子的吸引域是筛形域．在特定耦合参数区间中，解析证明这两个同步混沌吸引子的吸引域不仅被无穷远吸引子的吸引域筛形，还通过数值证明它们的吸引域彼此互相筛形，展示出类似于Wada性质的特征．但进一步的讨论表明这种复杂的被两个(或更多)吸引域共同筛形的结构并不是Wada域，而是由于筛形分岔和筛形域局部—全局分岔导致的．     

轴承-转子系统Hopf分叉及其后动力学行为研究

采用长轴承解析模型研究滑动轴承支承的平衡单盘柔性转子-轴承系统的自激振动,把结合打靶法的延续算法应用于柔性平衡转子-轴承系统Hopf分叉后周期解的追踪和求解上,基于Floquet理论对周期解的稳定性加以分析.通过持续追踪周期解频率变化并与失稳固有频率进行对比,分析了自激锁相现象,研究了非线性油膜力自激源对系统的作用机理.运用Poincare映射、分叉图、及Lyapnov指数对周期解分叉、混沌及进入和脱离混沌的过程进行了分析.     

两级悬浮EMS型磁悬浮控制系统的非线性动力学特性

在考虑二级悬浮弹簧的非线性特性的基础上，建立了两级悬浮EMS型磁悬浮控制系统的非线性动力学模型，给出了控制参数G1，G2的稳定性条件，进一步讨论了该磁悬浮系统在外界激励下的分叉行为及混沌动力特性，并利用Poincare映射，功率谱分析及最大Lyapunov指数等混沌运动的统计特征描述了该状态下控制系统的混沌运动特性。     

碰撞振动系统中周期轨擦边诱导的混沌激变

基于图胞映射理论, 提出了一种擦边流形的数值逼近方法, 研究了典型Du ng 碰撞振动系统中擦边诱导激变的全局动力学. 研究表明, 周期轨的擦边导致的奇异性使得系统同时产生1 个周期鞍和1 个混沌鞍. 当该周期鞍的稳定流形与不稳定流形发生相切时, 边界激变发生使得该混沌鞍演化为混沌吸引子. 噪声可以诱导周期吸引子发生擦边, 这种擦边导致了1 种内部激变的发生, 表现为该周期吸引子与其吸引盆内部的混沌鞍发生碰撞后演变为1 个混沌吸引子.     

二维耦合混沌同步系统的筛形品质因子

本文研究了在二维耦合混沌同步系统的混沌吸引子的筛形吸引域中,筛形品质因子与筛形吸引域的不确定指数之间存在着的联系,并通过由线性耦合达到混沌同步的标准帐篷映射系统的模型给出了数值例证。由于不确定指数接近于零,在一定的计算精度下,筛形品质因子是一定值。同时讨论了用筛形品质因子描述以筛形吸引域中的点为初始点的轨道被其混沌吸引子排斥的平均程序的合理性。     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

Chen系统的非线性动力学究研

利用受控Chen系统，并基于镜像操作方法，发现Chen吸引子是由左、右两个吸引子所组成的复合结构，且左、右吸引子均可由极限环生成。采用一维时间序列相空间重构技术和系统混沌的定量判据准则，揭示出Chen系统从规则运动转化到混沌运动所具有的普适特征：Chen系统可通过Pomeau-Manneville途径走向混沌，且其间歇性与Hopf分岔和倍周期分岔有关、在这些途径上既可观察到锁相和准周期运动，也可观察到类Chen吸引子、Chen系统和Lorenz系统之间的过渡吸引子和类Lorenz吸引子。     

Observing Symmetry-Breaking and Chaos in the Normal Form Network

The complex dynamical behaviors of neural networks may deducenew information processing methodology. In this paper, the dynamics of anormal form network with Z ₂ symmetry is studied. Thesecondary Hopf bifurcation of the network is discussed and a two-torusis observed. Examining the phase-locking motions of the two-torus, wepresent the regularity of symmetry-breaking occurring in the system. Ifthe ratio of the two frequencies of the codimension-two Hopf bifurcationis represented by an irreducible fraction, symmetry-breaking occurs wheneither the numerator or the denominator of the fraction is even. Chaoticattractors may be created with sigmoid nonlinearities added to theright-hand side of the normal form equations. The trajectory andsecond-order Poincaré maps of the chaotic attractor are given.The chaotic attractor looks like a butterfly on some of the second-orderPoincaré maps. This is a marvelous example for chaos mimickingnature.     

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible. We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle?CTakens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime. Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.     

Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays

This paper deals with dynamic behaviors on Hopfield type of ring neural network of four neurons having a pair of short-cut connections with multiple time delays. By suitable transformation and under certain assumptions on multiple time delays, the model is reduced to four dimensional nonlinear delay differential equations with three delays. Regarding these time delays as parameters, delay independent sufficient conditions for no stability switches of the trivial equilibrium of the linearized system are derived. Conditions for stability switching with respect to one delay parameter which is not associated with short-cut connection are obtained. Hopf bifurcations with respect to two other delays which are associated with short-cut connection are also obtained. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Using numerical simulations of the nonlinear model, different rich dynamical behaviors such as quasiperiodicity, torus attractor and chaotic-bands are also observed for suitable range of three delay parameters. Lyapunov exponents are also calculated using the AnT 4.669 tool for verification of chaotic dynamics.     

单调激活函数惯性项神经耦合系统的混沌共存

混沌及其稳态共存是神经网络系统中一个重要研究热点问题．本文基于惯性项神经元模型,利用非线性单调激活函数构造了一个惯性项神经耦合系统,采用理论分析和数值模拟相结合的方法,研究了系统平衡点以及静态分岔的类型,分析了系统两种不同模式的混沌及其稳态共存．具体来说,我们通过选取不同的初始值,利用相应的相位图和时间历程图,展现了系统混沌对初值的敏感依赖性．进一步,采用耦合强度作为动力学的分岔参数,研究了混沌产生的倍周期分岔机制,得到了单调激活函数耦合下的惯性项神经元系统混沌共存现象．     

Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures

This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.     

Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed.     

碰撞振动系统的一类余维二分岔及T2环面分岔

建立了三自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在两对共轭复特征值同时在单位圆上时,通过中心流形-范式方法将六维映射转变为四维范式映射.理论分析了这种余维二分岔问题,给出了局部动力学行为的两参数开折.证明系统在一定的参数组合下,存在稳定的Hopf分岔和T2环面分岔.数值计算验证了理论结果.     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

Dynamical analysis of two coupled parametrically excited van der Pol oscillators

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.