一类平面D3等变映射的混沌吸引子对称增加分歧的数值确定

本文讨论了一类平面D3等变映射的分歧和混沌性质。通过计算显示出映射随着参数的变化，从周期解走向混沌以及混沌吸引子由Z2-对称走向D3-对称的全过程。给出计算混沌吸引子的对称增加分歧扩张系统的算法，数值结果表明，两者相符。     

Couette-Taylor流三模系统的混沌行为及其仿真

研究了旋流式Couette-Taylor流三模态类Lorenz系统的动力学行为及其数值仿真问题.给出了此系统平衡点存在的条件,证明了其吸引子的存在性,给出了吸引子的Hausdorff维数上界的估计,数值模拟了系统分歧和混沌等的动力学行为发生的全过程,基于分岔图与最大Lyapunov指数谱和庞加莱截面以及功率谱和返回映射等仿真结果揭示了此系统混沌行为的普适特征.     

粉末注射成形动力方程的混沌特征分析及研究

粉末注射成形（PIM）充模过程具有混沌的特征．本文利用软件ANSYS对PIM充模过程的动力学方程数值求解。通过吸引子形态描述法分析对比两种不同注射速度下的动态吸引子图,计算混沌吸引子形态特征量,定量的表述混沌吸引子变化规律．研究表明粉末注射成形充模过程中存在混沌现象．     

(D6,Z2)-等变分歧问题的识别

利用奇点理论中光滑映射芽的接触等价，研究状态变量和分歧参数均具有对称性的等变分歧问题，给出了状态变量具有D。对称性，分歧参数具有Z2对称性的等变分歧问题的两个识别条件．     

D6等变非线性分歧问题的计算

本文研究D6对称群的分歧子群的分类,由此得到相应对称破缺分歧的计算方法,并应用到D6等变的Brusselator反应问题。     

一类双面碰撞振子的对称性尖点分岔与混沌

讨论了一类单自由度双面碰撞振子的对称型周期n-2运动以及非对称型周期n-2运动．把映射不动点的分岔理论运用到该模型,并通过分析对称系统的Poincaré映射的对称性,证明了对称型周期运动只能发生音叉分岔．数值模拟表明:对称系统的对称型周期n-2运动,首先由一条对称周期轨道通过音叉分岔形成具有相同稳定性的两条反对称的周期轨道;随着参数的持续变化,两条反对称的周期轨道经历两个同步的周期倍化序列各自生成一个反对称的混沌吸引子．如果对称系统演变为非对称系统,非对称型周期n-2运动的分岔过程可用一个两参数开折的尖点分岔描述,音叉分岔将会演变为一支没有分岔的分支以及另外一个鞍结分岔的分支．     

具有进位吸引子的区间映射

对于一个具有n-进位吸引子的区间连续映射,证明了：“n不是2的方幂”是该映射具有正拓扑熵的充分条件但不是必要条件；探讨了函数方程f~3(λx)=λf(x)的一类解,并证明这类解中的每一个成员都有3-进位吸引子．     

一类基于排斥吸引函数的离散混沌模型

利用排斥吸引函数容易实现有界离散映射的特点,提出了一类新的离散混沌映射,并利用理论推导对其分岔机理进行了研究.通过分岔图和Lyapunov指数谱清楚地展示了这个离散映射从有序到混沌的变化过程.可以看到这个简单的排斥吸引函数也能够象著名的Logistic模型一样产生复杂的动力学行为.     

Conformable分数阶单机无穷大电力系统分岔与混沌研究

基于Conformable分数阶微分定义和Adomian分解算法,设计了Conformable分数阶非线性系统半解析解算法和Lyapunov指数谱算法.采用Lyapunov指数谱、分岔图和吸引子相图分析了Conformable分数阶单机无穷大电力系统中的分岔与混沌现象,揭示了系统状态随参数和微分阶数变化时的规律以及系统走向混沌的道路.Matlab仿真数值模拟结果表明:Conformable分数阶单机无穷大电力系统的动力学特征丰富,系统产生混沌的最小阶数为0.41,系统初值的改变直接影响系统状态,并发现了多涡卷混沌吸引子和共存吸引子,功角失稳是产生多涡卷吸引子的根本原因.研究结果表明了求解算法的有效性与Conformable分数阶单机无穷大电力系统动力学特性的丰富性.     

一类非线性混沌系统混沌吸引子的冲击控制

在国内外研究工作的基础上,给出了一类非线性混沌系统混沌吸引子的冲击控制方案,运用普适方程的冲击控制理论导出了这类混沌系统混沌吸引子的冲击控制渐进稳定的条件,利用这一条件给出了混沌吸引子渐进稳定冲击控制的区间上界,最后给出了许多数据结果,这些结果对于混沌吸引子的控制将有重要的参考价值．     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Chaotic attractors in nonlinear dissipative systems of ordinary differential equations

A mechanism is proposed describing the formation of irregular attractors in a wide class of three-dimensional nonlinear autonomous dissipative systems of ordinary differential equations with singular cycles. The attractors of such systems, called singular attractors, lie on two-dimensional surfaces in the phase space and have no positive Lyapunov exponents. In all systems of this class the onset of chaos follows the same universal mechanism: a cascade of Feigenbaum’s period doubling bifurcations, a subharmonic cascade of Sharkovskii’s bifurcations, and eventually a homoclinic cascade. All classical chaotic systems, including Lorenz, Rössler, and Chua systems, satisfy these conditions.     

On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations

This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Bifurcation analysis and linear control of the Newton–Leipnik system

In this paper, we study a sort of chaotic system—Newton–Leipnik system which possesses two strange attractors. The static and dynamic bifurcations of the system are studied. The chaos controlling is performed by a simpler linear controller, and numerical simulation of the control is supplied. At the same time, Lyapunov exponents of the system show that the result of the chaos controlling is right.     

Effects of random noise in a dynamical model of love

This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of “Romeo’s romantic style”, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed.     

Characterization of chaotic attractors at bifurcations in Murali–Lakshmanan–Chua's circuit and one-way coupled map lattice system

In the present paper we study certain characteristic features associated with bifurcations of chaos in a finite dimensional dynamical system – Murali–Lakshmanan–Chua (MLC) circuit equation and an infinite dimensional dynamical system – one-way coupled map lattice (OCML) system. We characterize chaotic attractors at various bifurcations in terms of σ_n(q) – the variance of fluctuations of coarse-grained local expansion rates of nearby orbits. For all chaotic attractors the σ_n(q) versus q plot exhibits a peak at q=q_α. Additional peaks, however, are found only just before and just after the bifurcations of chaos. We show power-law variation of maximal Lyapunov exponent near intermittency and sudden widening bifurcations. Linear variation is observed for band-merging bifurcation. We characterize weak and strong chaos using probability distribution of k-step difference of a state variable.     

Effective approaches to explore rich dynamics of delay-differential equations

Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

On grazing and strange attractors fragmentation in non-smooth dynamical systems

This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-smooth dynamic systems. The sufficient and necessary conditions for grazing bifurcations in non-smooth dynamic systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-smooth dynamical systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.