两个周期激励下Duffing-van der Pol系统的混沌瞬态和广义激变

运用广义胞映射图方法研究两个周期激励作用下Duffing-van der Pol系统的全局特性.发现了系统的混沌瞬态以及两种不同形式的瞬态边界激变, 揭示了吸引域和边界不连续变化的原因. 瞬态边界激变是由吸引域内部或边界上的混沌鞍和分形边界上周期鞍的稳定流形碰撞产生.第一种瞬态边界激变导致吸引域突然变小, 吸引域边界突然变大; 第二种瞬态边界激变使两个不同的吸引域边界合并成一体.此外, 在瞬态合并激变中两个混沌鞍发生合并, 最后系统的混沌瞬态在内部激变中消失. 这些广义激变现象对混沌瞬态的研究具有重要意义. 关键词： 广义胞映射图方法 Duffing-van der Pol 混沌瞬态 广义激变     

一类含非黏滞阻尼的Duffing单边碰撞系统的激变研究

以一类含非黏滞阻尼的Duffing单边碰撞系统为研究对象, 运用复合胞坐标系方法, 分析了该系统的全局分岔特性. 对于非黏滞阻尼模型而言, 它与物体运动速度的时间历程相关, 能更真实地反映出结构材料的能量耗散现象. 研究发现, 随着阻尼系数、松弛参数及恢复系数的变化, 系统发生两类激变现象: 一种是混沌吸引子与其吸引域内的混沌鞍发生碰撞而产生的内部激变, 另一种是混沌吸引子与吸引域边界上的周期鞍(混沌鞍)发生碰撞而产生的常规边界激变(混沌边界激变), 这两类激变都使得混沌吸引子的形状发生突然改变. 关键词： 非黏滞阻尼 Duffing碰撞振动系统 激变 复合胞坐标系方法     

常微分方程系统中内部激变现象的研究

应用广义胞映射图论方法研究常微分方程系统的激变.揭示了边界激变是由于混沌吸引子与 在其吸引域边界上的周期鞍碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混 沌吸引子连同它的吸引域突然消失,在相空间原混沌吸引子的位置上留下了一个混沌鞍.研 究混沌吸引子大小(尺寸和形状)的突然变化,即内部激变.发现这种混沌吸引子大小的突然 变化是由于混沌吸引子与在其吸引域内部的混沌鞍碰撞产生的,这个混沌鞍是相空间非吸引 的不变集,代表内部激变后混沌吸引子新增的一部分.同时研究了这个混沌鞍的形成与演化. 给出了对永久自循环胞集和瞬态自循环胞集进行局部细化的方法. 关键词： 广义胞映射 有向图 激变 混沌鞍     

一类新的边界激变现象:混沌的边界激变

混沌吸引子的激变是一类普遍现象.借助于广义胞映射图论(generalized cell mapping digraph)方法发现了嵌入在分形吸引域边界内的混沌鞍,这个混沌鞍由于碰撞混沌吸引子导致混沌吸引子完全突然消失,是一类新的边界激变现象,称为混沌的边界激变.可以证明混沌的边界激变是由于混沌吸引子与分形吸引域边界上的混沌鞍相碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混沌吸引子连同它的吸引域突然消失,同时这个混沌鞍也突然增大 关键词： 广义胞映射 有向图 激变 混沌鞍     

混沌吸引子的对称破缺激变

许多非线性动力系统都有某种对称性,在不同情形下可有不同的表现形式,但始终保持其对称的特点.不同对称形式间的转变导致对称破缺分岔或激变.关于非线性动力系统中相空间运动轨道的对称破缺分岔,已有大量研究工作,但绝大多数是指周期或拟周期相轨的对称破缺,偶尔提到对称系统中的混沌相轨也存在“对偶性”.最近,在简谐外激Duffing系统周期轨道对称破缺引发鞍-结分岔的研究中,得到了分岔后由Poincaré映射点间断流构成的图像,其中包括两个稳定周期结点、一个周期鞍点,及其稳定流形与不稳定流形,均较规则.本工作研究了正弦 关键词： 对称破缺 混沌 激变 分形吸引域     

非自治分数阶Duffing系统的激变现象

对一个非自治分数阶Duffing系统的激变现象进行了研究.首先介绍了一种研究分数阶非线性系统全局动力学的数值方法,即拓展的广义胞映射方法 (EGCM).该方法是基于分数阶导数的短记忆原理,并结合了广义胞映射方法和改进的预估校正算法,根据胞空间的特点,将胞尺寸作为截断误差的参考值,以此得到了一步映射时间的估算公式.用EGCM方法分别研究了分数阶Duffing系统随分数阶导数的阶数和外激励强度变化发生的边界激变和内部激变.并基于此,将激变拓展定义为混沌基本集与周期基本集之间的碰撞,其中混沌基本集包括混沌吸引子,边界上的混沌集合以及吸引域内部的非混沌吸引子的混沌集合.所得结果进一步说明了EGCM方法对于分析分数阶系统全局动力学的有效性.     

Hénon映射的噪声诱导间歇现象及其临界值估算研究

本文研究了Hénon映射在噪声诱导下发生的间歇现象.通过数值模拟和全局分析手段,揭示了噪声诱导间歇现象的机理.基于随机敏感度函数法,通过检测噪声作用下周期吸引子的置信椭圆与混沌鞍的碰撞情况,给出了诱发间歇现象的噪声强度临界值的估算方法.结果表明,Hénon映射中噪声诱导间歇现象是由随机周期吸引子和混沌鞍不稳定流形的相互作用引发,随机敏感度函数的方法可以较好地估算发生间歇现象的噪声强度临界值.     

两参量平面上双重激变尖点研究

应用广义胞映射图论(GCMD)方法,研究两参量正弦强迫振子的双重激变现象,确定了两参量平面上的双重激变尖点,在这个尖点上两条边界激变曲线和两条内部激变曲线相汇交,四种不同的激变重合.物理上,在这样一个尖点附近的参量扰动(噪声)导致动力学行为戏剧性变化. 关键词： 全局分析 广义胞映射 双重激变尖点 混沌鞍     

一个强、弱耗散功能可分隔系统中的特征激变

报道一个由保守映象和耗散映象不连续、不可逆地分段描述的系统，以及在其中发生的一例特征激变.激变的独特之处在于逃逸孔洞.由映象的不连续、不可逆性而导致相平面中出现一个胖分形迭代禁区网，它使得一个混沌吸引子突然失稳而发生激变后出现的两个周期吸引子的吸引域边界成为点滴状.仅仅在每个周期点邻近存在这样的一个作为逃逸孔洞的、受到强耗散性支配和禁区边界限制的规则边界吸引域. 关键词： 激变 保守映象 耗散映象 逃逸孔洞     

形状记忆合金薄板系统全局激变现象分析

形状记忆合金在工程应用中的难点主要来自于系统在温度和外载荷作用下产生的复杂全局动力学行为.本文以形状记忆合金薄板动力系统为研究对象,分析在温度和激励振幅两个控制参数作用下系统的全局动力学.通过全局分岔图,可以观测到系统会发生复杂的激变现象,然后利用复合胞坐标系方法,获取系统的吸引子、吸引域、鞍和域边界等信息,展现系统的全局演变过程.研究发现,系统随着振幅和温度变化会呈现复杂的全局结构,并发生一系列的边界激变、合并激变现象,同时多次发生分形-Wada, Wada-Wada, Wada-分形等域边界突变.通过对指定区域细化,可以清晰地显示域边界的分形特征.研究结果对于如何通过调控温度与外载荷强度,使形状记忆合金薄板在系统中发挥最佳性能具有理论指导意义.     

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

Wada boundary bifurcations induced by boundary saddle–saddle collision

In this letter, a Wada boundary bifurcation (WBB) induced by a boundary saddle touching another boundary saddle is first found through the study of a forced damped pendulum. The WBB can be quantitatively described by the change both in the number of basins involved and in the geometrical size of the boundary. We perceive the manifold structures of the two saddles, that is, a pre-existence of heteroclinic crossing and the other nearly forming heteroclinic tangency exist before the WBB. So we schematically construct the equivalent topological structure of the manifolds of arbitrary two saddles, and rigorously prove two theorems that indicate the existence of the heteroclinic tangency and thus generically confirm the mechanism of such WBB.     

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.     

A fuzzy crisis in a Duffing-van der Pol system

A crisis in a Duffing--van del Pol system with fuzzy uncertainties is studied by means of the fuzzy generalised cell mapping (FGCM) method. A crisis happens when two fuzzy attractors collide simultaneously with a fuzzy saddle on the basin boundary as the intensity of fuzzy noise reaches a critical point. The two fuzzy attractors merge discontinuously to form one large fuzzy attractor after a crisis. A fuzzy attractor is characterized by its global topology and membership function. A fuzzy saddle with a complicated pattern of several disjoint segments is observed in phase space. It leads to a discontinuous merging crisis of fuzzy attractors. We illustrate this crisis event by considering a fixed point under additive and multiplicative fuzzy noise. Such a crisis is fuzzy noise-induced effects which cannot be seen in deterministic systems.     

Chaotic transients in spatially extended systems

Different transient-chaos related phenomena of spatiotemporal systems are reviewed. Special attention is paid to cases where spatiotemporal chaos appears in the form of chaotic transients only. The asymptotic state is then spatially regular. In systems of completely different origins, ranging from fluid dynamics to chemistry and biology, the average lifetimes of these spatiotemporal transients are found, however, to grow rapidly with the system size, often in an exponential fashion. For sufficiently large spatial extension, the lifetime might turn out to be larger than any physically realizable time. There is increasing numerical and experimental evidence that in many systems such transients mask the real attractors. Attractors may then not be relevant to certain types of spatiotemporal chaos, or turbulence. The observable dynamics is governed typically by a high-dimensional chaotic saddle. We review the origin of exponential scaling of the transient lifetime with the system size, and compare this with a similar scaling with system parameters known in low-dimensional problems. The effect of weak noise on such supertransients is discussed. Different crisis phenomena of spatiotemporal systems are presented and fractal properties of the chaotic saddles underlying high-dimensional supertransients are discussed. The recent discovery according to which turbulence in pipe flows is a very long lasting transient sheds new light on chaotic transients in other spatially extended systems.     

Boundary crisis and transient in a dissipative relativistic standard map

Some dynamical properties for a problem concerning the acceleration of particles in a wave packet are studied. The model is described in terms of a two-dimensional nonlinear map obtained from a Hamiltonian which describes the motion of a relativistic standard map. The phase space is mixed in the sense that there are regular and chaotic regions coexisting. When dissipation is introduced, the property of area preservation is broken and attractors emerge. We have shown that a tiny increase of the dissipation causes a change in the phase space. A chaotic attractor as well as its basin of attraction are destroyed thereby leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with the stable manifold of a saddle fixed point. Once the chaotic attractor is destroyed, a chaotic transient described by a power law with exponent −1 is observed.