非线性动力系统分岔图中的暗线

基于非线性动力系统混沌运动的回归特性，构造了一种对分岔图中穿过混沌区的暗线进行研究的数值回归算法。运用该算法求得抛物线映射的暗线，并与通过暗线方程精确求得的暗线进行比较，验证了算法的有效性。对Brussel振子系统和分段线性单级齿轮动力系统的暗线进行了研究。通过对非线性动力系统分岔图中暗线的研究，由其切点可以得到嵌在混沌区中的周期窗口，由其交点可以得到混沌吸引子的激变点。研究结果表明该算法有助于分析系统的动力学行为和控制混沌运动。     

五维对流模式吸引子的Lyapunov谱

利用Lyapunov谱的算法,计算了五维对流非线性动力系统吸引子的Lyapunov谱及奇怪吸引子的维数。     

二维正弦离散映射的分岔和吸引子

由一个正弦映射和一个三次方映射通过非线性耦合,构成一个新的二维正弦离散映射. 基于此二维正弦离散映射得到系统的不动点以及相应的特征值,分析了系统的稳定性,研究了系统的复杂非线性动力学行为及其吸引子的演变过程. 研究结果表明：此二维正弦离散映射中存在复杂的对称性破缺分岔、Hopf分岔、倍周期分岔和周期振荡快慢效应等非线性物理现象. 进一步根据控制变量变化时系统的分岔图、Lyapunov指数图和相轨迹图分析了系统的分岔模式共存、快慢周期振荡及其吸引子的演变过程,通过数值仿真验证了理论分析的正确性. 关键词： 正弦离散映射 对称性破缺分岔 Hopf分岔 吸引子     

一类具有两个边界的分段光滑系统中边界碰撞分岔现象及混沌

以不连续运行模式下的电流反馈型Buck-Boost变换器为例，导出了一类具有三段形式的分段光滑迭代映射方程，数值仿真得到了输入电压变化时的分岔图.结果表明，发生分岔时映射雅可比矩阵的特征值以不连续的方式跳跃出复平面上的单位圆，而且映射总有某个或某些轨道点位于相平面中不同区域的边界上，即映射随输入电压的变化会发生边界碰撞分岔现象，如由周期态到周期态以及由周期态到混沌态的分岔. 关键词： 分段光滑系统 边界碰撞分岔 混沌     

状态反馈和参数调整控制离散非线性系统的倍周期分岔和混沌

利用系统的状态反馈和参数调节的方法，有效地实现了离散非线性动力系统的倍周期分岔的延迟控制和混沌吸引子中不稳定周期轨道的控制.同时，通过选择合适的调控参数，可以将系统从一个2n周期轨道控制到2m(m关键词： 状态反馈 参量调节 混沌控制 分岔控制     

相对转动非线性动力系统的稳定性与强迫激励下的近似解

研究相对转动非线性动力系统的运动稳定性.建立具有一般广义阻尼力和外扰激励的一类两质量相对转动非线性动力系统的动力学方程.研究相对转动非线性动力自治系统的稳定性，证明系统在一定条件下可发生闭轨分岔.应用多尺度法得到强迫激励下非自治系统的近似解. 关键词： 相对转动 非线性动力系统 运动稳定性 近似解     

一维Tonks-Girardeau原子气区域中 Gross-Pitaevskii方程简化模型的精确行波解

利用动力系统方法研究一维Tonks-Girardeau原子气区域中Gross-Pitaevskii (GP)方程简化模型的一些精确行波解以及这些精确行波解的动力学行为, 研究系统的参数对行波解的动力学行为的影响. 在不同的参数条件下, 获得了一维Tonks-Girardeau原子气区域中GP方程简化模型的六个行波解的精确参数表达式. 关键词： 动力系统方法 孤立波解 周期波解 扭波解     

非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

非线性传送带系统的复杂分岔

讨论了一类单自由度非线性传送带系统. 首先通过分段光滑动力系统理论得出系统滑动区域的解析分析和平衡点存在性条件; 其次利用数值方法, 对系统几种类型的周期轨道进行单参数和双参数延拓, 得到系统的余维一滑动分岔曲线和若干余维二滑动分岔点, 以及系统在参数空间中的全局分岔图. 通过对系统分岔行为的研究, 反映出传送带速度和摩擦力振幅对系统动力学行为有较大影响, 揭示了非线性传送带系统的复杂动力学现象. 关键词： 传送带系统 滑动分岔 周期运动     

研究强非线性振动系统同宿分岔问题的规范形方法

以改进的规范形理论为基础,采用强非线性振动问题的分析方法,拓展了原有弱非线性振动系统同宿分岔判据的适用范围.首先在复规范形求解过程中引入待定固有频率,计算了一类单自由度强非线性振动系统的周期解.然后分别依据系统的待定固有频率趋于零和周期轨道趋近于鞍点两条途径获得了强非线性振动条件下系统同宿分岔的解析判据.最后通过与原有解析结果和数值结果相比较验证了本文方法的有效性. 关键词： 规范形 同宿分岔 强非线性 周期解     

Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example.     

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.     

A new hyperchaos system and its circuit simulation by EWB

This paper reports a new hyperchaotic system evolved from the three-dimensional Lü chaotic system. The Lyapunov exponents spectrum and the bifurcation diagram of this new hyperchaotic system are obtained. Hyperchaotic attractor, periodic orbit and chaotic attractor are obtained by computer simulation. A circuit is designed to realize this new hyperchaotic system by electronic workbench.     

Dynamics of a new composite four–Scroll chaotic system

A new three-dimensional autonomous chaotic system is proposed. This new system can generate single-scroll, double-scroll, three-scroll and four-scroll attractors under different system parameters. Particularly, it can generate a four-scroll chaotic attractor composed of a large Chua-like attractor and a small Lorenz-like attractor. And the system can also generate a nested three-scroll attractor and the multi-double-scroll chaotic attractor. In addition, the system possesses the chaotic state transition, and the number of scrolls will change in the state transition process. The formation mechanism of the composite four-scroll chaotic attractor is analyzed in detail. The dynamic analysis methods include time series, 0–1 test chart, phase diagram, bifurcation diagram and Lyapunov exponents are used to describe some basic dynamics behaviors of the proposed system.     

A new hyperchaotic system and its linear feedback control

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter k. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.     

共轭Chen混沌系统的分岔分析及基于该系统的超混沌生成研究

分析了一个三维自治混沌系统的Hopf分岔现象,该系统的混沌吸引子属于共轭Chen混沌系统.通过引入一个控制器,基于该混沌系统构建了一个四维自治超混沌系统.该超混沌系统含有一个单参数,在一定的参数范围内呈现超混沌现象.通过Lyapunov指数和分岔分析,随着参数的变化该系统轨道呈现周期轨道、准周期轨道、混沌和超混沌的演化过程. 关键词： 混沌 超混沌生成 Hopf分岔 分岔分析     

On the complexity of periodic and nonperiodic behaviors of a hysteresis-based electronic oscillator

We investigate the families of periodic and nonperiodic behaviors admitted by a hysteresis-based circuit oscillator. The analysis is carried out by combining brute-force simulations with continuation methods. As a result of the analysis, it is shown that the existence of many different periodic solutions and of the chaotic behaviors associated with them is organized by few codimension-2 bifurcation points. This implies the possibility of switching between different periodic solutions by controlling only two bifurcation parameters, which makes the oscillator a possible generator of nontrivial periodic solutions suitable, for instance for actual radiofrequency identification systems applications.     

基于参数切换算法的混沌系统吸引子近似及其电路设计

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.     

A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system

This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms, and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis, the Hopf bifurcation processes are proved to arise at certain equilibrium points. Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours; the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally, an analog electronic circuit is designed to physically realize the chaotic system; the existence of four-wing chaotic attractor is verified by the analog circuit realization.     

A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.