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。     

Wada分形吸引域边界上的混沌鞍

应用广义胞映射图论（Generalized Cell Mapping Digraph)方法，数值地研究Thompson的逃逸方程在最佳逃逸点附近的分岔。发现了嵌入在Wada分形吸引域边界上的混沌鞍，混沌鞍是状态空间不稳定（非吸引）的混沌不变集合。Wada分形吸引域边界是具有Wada性质的边界，即吸引域边界上的任意点也同时是至少两个其它吸引域的边界点，称为Wada域边界。我们证明Wada域边界上的混沌鞍导致局部鞍结分岔具有全局不确定性结局，研究了Wada域边界上混沌鞍的形成与演化，证明最终的逃逸分岔是混沌吸引子碰撞混沌鞍的边界激变。     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

Duffing-van der Pol系统的随机分岔

应用广义胞映射图论方法(GCMD)研究了在谐和激励与随机噪声共同作用下的Duffing-van der Pol系统的随机分岔现象. 系统参数选择在多个吸引子与混沌鞍共存的范围内. 研究发现, 随着随机激励强度的增大,该系统存在两种分岔现象: 一种为随机吸引子与吸引域边界上的鞍碰撞, 此时随机吸引子突然消失; 另一种为随机吸引子与吸引域内部的鞍碰撞, 此时随机吸引子突然增大. 研究证实, 当随机激励强度达到某一临界值时, 该系统还会发生D-分岔(基于Lyapunov指数符号的改变而定义), 此类分岔点不同于上述基于系统拓扑性质改变所得的分岔点.     

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.     

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．     

齿轮传动系统共存吸引子的不连续分岔

大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素.多吸引子共存时,运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换,对整个机器产生不良的影响.目前,一些隐藏的吸引子没有被发现,共存吸引子的分岔演化规律没有被完全揭示.考虑单自由度直齿圆柱齿轮传动系统,构建由局部映射复合的Poincaré映射,给出Jacobi矩阵特征值计算的半解析法.应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔,应用胞映射法计算共存吸引子的吸引域,讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响,揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为.倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞,使倍周期分岔呈现亚临界特性.鞍结分岔是共存周期吸引子出现或消失的主要原因.边界激变引起混沌吸引子及其吸引域突然消失,对应周期吸引子的分岔终止.     

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．     

离散达芬映射中由边界激变所诱发的复杂的张弛振荡

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2~n(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。     

A special type of codimension two bifurcation and unusual dynamics in a phase-modulated system with switched strategy

We investigate a new type of codimension two bifurcation and related dynamics in a phase-modulated system with switched strategy. Two curves intersect at a point and are called the crisis–Hopf bifurcation. At the critical crisis–Hopf vertex, a boundary crisis and Hopf bifurcation coincide. Metamorphosis of coexisting attractors can be observed in the vicinity of the vertex. Another novelty is that we discover some sets of measure zero with riddled holes in the neighborhood of the bifurcation point. These sets display some qualitative and quantitative features of riddled basins but they are essentially different from the riddled basin. It may provide a more comprehensive picture for unusual dynamical features.     

Chaotic Saddles in Wada Basin Boundaries and Their Bifurcations by the Generalized Cell-Mapping Digraph (GCMD) Method

By means of the generalized cell-mapping digraph (GCMD) method, we studybifurcations governing the escape of periodically forced oscillatorsfrom a potential well, in which a chaotic saddle plays an extremelyimportant role. In this paper, we find the chaotic saddle anddemonstrate that it is embedded in a strange fractalbasin boundary which has the Wada property that any point that is on theboundary of that basin is also simultaneously on the boundary of atleast two other basins. The chaotic saddle in the Wada basin boundary,by colliding with a chaotic attractor, leads to a chaotic boundarycrisis with indeterminate outcome. A local saddle-node fold bifurcation,if the saddle of the saddle-node fold is located in tangency with thechaotic saddle in the Wada basin boundary, also results in a strangeglobal phenomenon, namely that the local saddle-node fold bifurcation hasglobally indeterminate outcome. We also investigate the origin andevolution of the chaotic saddle in the Wada basin boundary, particularlyconcentrating on its discontinuous bifurcations (metamorphoses). Wedemonstrate that the chaotic saddle in the Wada basin boundary iscreated by a collision between two chaotic saddles in differentfractal basin boundaries. After a final escape bifurcation, there onlyexists the attractor at infinity and a chaotic saddle with a beautifulpattern is left behind in the phase space.     

Kawakami映射的超混沌行为研究

通过对一类平面二维映射系统非线性动力学行为的分析,发现该系统存在一个奇怪吸引子,该吸引子具有两个正Lyapunov指数和分数维。通过该系统不动点的分析揭示了该吸引子的吸引域边界结构,即不稳定第二类结点与不稳定偶数周期点在吸引域边界上的相间排列。     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System

The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.     

Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control

The nonlinear dynamics of a differential system describing the motion of a vehicle driven by a pilot is examined. In a first step, the stability of the system near the critical speed is analyzed by the bifurcation method in order to characterize its behavior after a loss of stability. It is shown that a Hopf bifurcation takes place, the stability of limit cycles depending mainly on the vehicle and pilot model parameters. In a second step, the front wheels of the vehicle are assumed to be subjected to a periodic disturbance. Chaotic and hyperchaotic motions are found to occur for some range of the speed parameter. Numerical simulations, such as bifurcation diagrams, Poincaré maps, Fourier spectrums, projection of trajectories, and Lyapunov exponents are used to establish the existence of chaotic attractors. Multiple attractors may coexist for some values of the speed, and basins of attraction for such attractors are shown to have fractal geometries.     

Fractional-order PWC systems without zero Lyapunov exponents

In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems.