耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

分段线性混沌电路的非光滑分岔分析

讨论了分段线性的电容混沌电路的动力学行为.由数值模拟得到了对称的周期解和混沌吸引子.通过引入广义Jacobian矩阵,以周期解为例,从理论上分析了系统由电容电量的分段线性而引起的非光滑分岔,并合理解释了系统动力学行为产生的机理及其演化规律,其结论与数值计算的结果大致符合.     

分段线性电路切换系统的复杂行为及非光滑分岔机理

分析了分段线性电路系统在周期切换下的复杂动力学行为及其产生的机理. 基于平衡点分析, 给出了两子系统Fold分岔和Hopf分岔条件. 考虑了在不同稳定态时两子系统周期切换的分岔特性, 产生了不同的周期振荡, 并揭示了其产生的机理. 在不同的周期振荡中, 切换点的数量随参数变化产生倍化, 导致切换系统由倍周期分岔进入混沌.关键词：分段线性电路切换系统非光滑分岔     

Noise destroys the coexistence of periodic orbits of a piecewise linear map

The effects of Gaussian white noise and Gaussian colored noise on the periodic orbits of period-5(P-5) and period-6(P-6) in their coexisting domain of a piecewise linear map are investigated numerically.The probability densities of some orbits are calculated.When the noise intensity is D = 0.0001,only the orbits of P-5 exist,and the coexisting phenomenon is destroyed.On the other hand,the self-correlation time τ of the colored noise also affects the coexisting phenomenon.When τcττc,only the orbits of P-5 appear,and the stability of the orbits of P-5 is enhanced.However,when ττc(τc and τc are critical values),only the orbits of P-6 exist,and the stability of the P-6 orbits is enhanced greatly.When ττc,the orbits of P-5 and P-6 coexist,but the stability of the P-5 orbits is enhanced and that of P-6 is weakened with τ increasing.     

Statistics of strange attractors by generalized cell mapping

It is proposed in this paper to use the generalized cell mapping to locate strange attractors of dynamical systems and to determine their statistical properties. The cell-to-cell mapping method is based upon the idea of replacing the state space continuum by a large collection of state space cells and of expressing the evolution of the dynamical system in terms of a cell-to-cell mapping. This leads to a Markov chain which in turn allows us to compute all the statistical properties as well as the invariant distribution. After a general discussion, the method is applied in this paper to strange attractors of a variety of systems governed either by point mappings or by differential equations. The results indicate that it is a viable, effective and attractive method. Some comments on this method in comparison with the method of direct iteration will also be made.     

Pruning-induced phase transition observed by a scattering method

In hyperbolic systems, transient chaos is associated with an underlying chaotic saddle in phase space. The structure of the chaotic saddle of a class of piecewise linear, area-preserving, two-dimensional maps with overall constant Lyapunov exponents has been observed by a scattering method. The free energy obtained in this way displays a phase transition at <0 in spite of the fact that no phase transition occurs in the free energy dedcued from the spectrum of Lyapunov exponents. This is possible because pruning introduces a second effective scaling exponent by creating, at each level of the approximation, particular small pieces in the incomplete Cantor set approximating the saddle. The second scaling arises for a subset of values of the control parameter that is dense in the parameter interval.     

On the symmetry-breaking bifurcation of chaotic attractors

A new type of crisis is shown to exist in a broad class of systems (including the Lorenz model) which leads to an anomalous band splitting or to a symmetry-breaking bifurcation of the strange attractor, depending on the actual values of the control parameters. A piecewise linear model is used to understand the mechanism of this crisis and to obtain exact results.     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

基于混沌神经网络的单向Hash函数

提出了一种基于混沌神经网络的单向Hash函数,该方法通过使用以混沌分段线性函数作为输出函数的神经网络和基于时空混沌的密钥生成函数实现明文和密钥信息的混淆和扩散,并基于密码块连接模式实现对任意长度的明文序列产生128位的Hash值.理论分析和实验结果表明,提出的Hash函数可满足所要求的单向性,初值和密钥敏感性,抗碰撞性和实时性等要求.关键词：混沌神经网络Hash函数分段线性混沌映射时空混沌     

Jerk系统耦合的分岔和混沌行为

通过分析耦合的Jerk系统的平衡点及其稳定性,给出了参数空间中不同的分岔集,进而将参数空间划分为对应于各种动力学行为的不同区域.探讨了耦合系统随不同参数变化的动力学演化过程,重点分析了系统耦合强度变化对其动力学行为的影响.揭示了多种运动模式共存及倍周期分岔等各种非线性现象的产生机理.     

一个全局耦合不连续映像格子中的冻结化随机图案模式

本文研究了一类既不连续又不可逆分段线性映像构成的全局耦合映像格子系统中的一类典型集体动力学行为, 即冻结化随机图案模式. 计算了平均同步序参量和最大李雅普诺夫指数随耦合强度的变化. 结果显示, 当耦合强度超过某个阈值后, 在给定动力学变量的初始下, 系统几乎都能达到完全或部分同步状态, 出现冻结化随机图案. 这些现象表明, 耦合映像格子系统中存在着多个共存的吸引子. 因此, 其冻结化图案的结构和分布敏感地依赖于格点动力学变量初始值的选取. 感兴趣地是, 即使当单映像处于混沌状态时, 格点间的耦合仍能将系统调制到规则的运动状态, 这种特征对于混沌控制具有重要的利用价值. 上述丰富动力学行为的出现是由于单映像中不连续性和不可逆性相互作用的结果.     

Truncated Navier-stokes equations: continuous transition from a five-mode to a seven-mode model

A two-parameter family of nonlinear differential equations x=F(x, R, ) is studied, which allows one to connect continuously, as varies from zero to one, the different phenomenologies exhibited by a model of 5-mode truncated Navier-Stokes equations and by a 7-mode one extending it. A critical value is found for, at which the most significant phenomena of the 5-mode system either vanish or go to infinity. New phenomena arise then, leading to the 7-mode model.Supported by G.N.F.M., C.N.R.     

Freudenthal duality and generalized special geometry

Freudenthal duality, introduced in Borsten et al. (2009) [1] and defined as an anti-involution on the dyonic charge vector in d=4 space-time dimensions for those dualities admitting a quartic invariant, is proved to be a symmetry not only of the classical Bekenstein-Hawking entropy but also of the critical points of the black hole potential.Furthermore, Freudenthal duality is extended to any generalized special geometry, thus encompassing all N>2 supergravities, as well as N=2 generic special geometry, not necessarily having a coset space structure.     

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     

Synchronization transition of a coupled system composed of neurons with coexisting behaviors near a Hopf bifurcation

The coexistence of a resting condition and period-1 firing near a subcritical Hopf bifurcation point, lying between the monostable resting condition and period-1 firing, is often observed in neurons of the central nervous systems. Near such a bifurcation point in the Morris-Lecar （ML） model, the attraction domain of the resting condition decreases while that of the coexisting period-1 firing increases as the bifurcation parameter value increases. With the increase of the coupling strength, and parameter and initial value dependent synchronization transition processes from non-synchronization to compete synchronization are simulated in two coupled ML neurons with coexisting behaviors： one neuron chosen as the resting condition and the other the coexisting period-1 firing. The complete synchronization is either a resting condition or period-1 firing dependent on the initial values of period-1 firing when the bifurcation parameter value is small or middle and is period- 1 firing when the parameter value is large. As the bifurcation parameter value increases, the probability of the initial values of a period- 1 firing neuron that lead to complete synchronization of period- 1 firing increases, while that leading to complete synchronization of the resting condition decreases. It shows that the attraction domain of a coexisting behavior is larger, the probability of initial values leading to complete synchronization of this behavior is higher. The bifurcations of the coupled system are investigated and discussed. The results reveal the complex dynamics of synchronization behaviors of the coupled system composed of neurons with the coexisting resting condition and period-1 firing, and are helpful to further identify the dynamics of the spatiotemporal behaviors of the central nervous system.