耦合帐篷映射混沌同步系统的筛形吸引域

讨论了两个线性耦合的标准帐篷映射混沌同步系统的同步混沌吸引子的吸引域,证明其是筛形吸引域.提出了筛形品质因子的概念,并据之给出了系统的同步混沌吸引子的吸引域发生 从局部筛形到全局筛形的转变的耦合参数临界值.修正了当系统出现筛形吸引域时的横截Lyapunov指数的解析表达式.指出考查筛形吸引域在混沌同步系统中有着重要意义. 关键词： 混沌同步 筛形吸引域 耦合帐篷映射     

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     

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     

一类四维混沌系统切换混沌同步

构建了一类可切换的四维混沌系统，通过选择器实现这类系统间的随机切换.简要地分析了四维混沌系统平衡点的性质、混沌吸引子的相图和Lyapunov指数等特性，并设计了实现四维混沌系统切换的实际电路.利用非线性反馈控制方法实现了这类系统与其中某个系统之间的切换混沌同步.根据系统稳定性理论，得到了非线性反馈控制器的结构和系统达到混沌同步时反馈控制增益的取值范围. 关键词： 非线性反馈 混沌同步 四维混沌系统     

阈值耦合混沌神经元的同步研究

将阈值控制方法应用于混沌神经元团簇,构成阈值耦合混沌神经元映射,研究其时空特性.仿真实验表明,控制阈值决定了阈值耦合混沌神经元映射输出的时间周期特性,而张弛时间影响了输出的空间特性,阈值耦合混沌神经元映射输出表现出很好的聚类性.当张弛时间足够大时,阈值耦合混沌神经元映射输出实现时空完全同步.     

广义Hénon映射的滑模变结构控制同步

基于可线性化的非线性离散变结构跟踪控制方法实现了广义Hénon映射混沌系统的同步.广义Hénon映射的混沌吸引子比Hénon映射的混沌吸引子要复杂得多,对于保密通信来说,这种复杂性正是所期望的.提出的同步方法允许参数有适当的失配程度,这对工程实现是非常有利的,理论分析和仿真结果证实了该方法的有效性 关键词： 广义Hénon映射 混沌 同步 变结构控制     

布拉格声光双稳系统时空混沌的单向耦合同步

使用非线性动力学中的一维和二维耦合格子模型研究两个声光双稳系统的时空混沌同步.将驱动系统的输出以适当的比例耦合到响应系统并进行均衡, 能实现两系统的时空混沌同步.利用计算最大条件Lyapunov指数, 给出达到同步所需的最小耦合强度与系统参数的关系. 数值实验表明,在小噪声影响时仍然可以实现两系统的同步, 此法具有一定的抗干扰能力. 关键词： 单向耦合同步 时空混沌 布拉格声光双稳系统     

连续时间混沌系统的自适应Ｈ∞ 同步方法

提出了一种连续时间混沌系统的自适应H∞同步方法.当响应系统没有受噪声干扰影 响时，所设计的自适应H∞同步控制器能够使响应系统与驱动系统全局同步.当响应 系统受噪声干扰影响时，自适应H∞同步控制器能够使噪声干扰对同步误差的影响小于期望的水平.最后，以蔡氏电路混沌系统为例来说明所提方法的有效性. 关键词： 混沌 混沌同步 自适应控制 H∞控制 自适应H∞同步     

非线性系统中的关联色噪声

研究了加性噪声和乘性噪声之间的耦合为色噪声时非线性系统的动力学行为.对于不同的噪声关联时间τ,求出了系统的有效本征值λeff和强度相关时间Tc.结果表明,当噪声之间的耦合λ大于零时,关联时间τ的增大可抑制系统在阈值附近的涨落;而当噪声之间的耦合λ小于零时,关联时间τ的增大则加强系统在阈值附近的涨落 关键词： 耦合强度 噪声关联时间 有效本征值 强度相关时间     

Sprott-B和Sprott-C系统之间的耦合混沌同步

Sprott-B和 Sprott-C是拓扑等价的异结构混沌系统. 利用系统变量线性耦合方法实现二系统之间的混沌同步；根据Lyapunov稳定性理论，确定了二系统达到同步时耦合系数的阈值. 设计了实现二系统耦合同步的实验电路，实验结果证明了理论分析的正确性. 基于Sprott-B和 Sprott-C异结构系统混沌同步，提出了一种新的具有更好保密性能的混沌保密通讯方法. 关键词： 线性耦合 混沌同步 异结构系统 拓扑等价     

The isolated critical value phenomenon in local－global riddling bifurcation

A chaotic synchronized system of two coupled skew tent maps is discussed in this paper. The locally and globally riddled basins of the chaotic synchronized attractor are studied. It is found that there is a novel phenomenon in the local－global riddling bifurcation of the attractive basin of the chaotic synchronized attractor in some specific coupling intervals. The coupling parameter corresponding to the locally riddled basin has a single value which is embedded in the coupling parameter interval corresponding to the globally riddled basin, just like a breakpoint. Also, there is no relation between this phenomenon and the form of the chaotic synchronized attractor. This phenomenon is found analytically. We also try to explain it in a physical sense. It may be that the chaotic synchronized attractor is in the critical state, as it is infinitely close to the boundary of its attractive basin. We conjecture that this isolated critical value phenomenon will be common in a system with a chaotic attractor in the critical state, in spite of the system being discrete or differential.     

Multifractal structure of a riddled basin

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.     

Riddling and chaotic synchronization of coupled piecewise-linear Lorenz maps

We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations, as previously suggested by numerical experiments. The use of piecewise-linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins.     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

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.     

Chaos and generalised multistability in a mesoscopic model of the electroencephalogram

We present evidence for chaos and generalised multistability in a mesoscopic model of the electroencephalogram (EEG). Two limit cycle attractors and one chaotic attractor were found to coexist in a two-dimensional plane of the ten-dimensional volume of initial conditions. The chaotic attractor was found to have a moderate value of the largest Lyapunov exponent (3.4 s⁻¹ base e) with an associated Kaplan-Yorke (Lyapunov) dimension of 2.086. There are two different limit cycles appearing in conjunction with this particular chaotic attractor: one multiperiodic low amplitude limit cycle whose largest spectral peak is within the alpha band (8-13 Hz) of the EEG; and another multiperiodic large-amplitude limit cycle which may correspond to epilepsy. The cause of the coexistence of these structures is explained with a one-parameter bifurcation analysis. Each attractor has a basin of differing complexity: the large-amplitude limit cycle has a basin relatively uncomplicated in its structure while the small-amplitude limit cycle and chaotic attractor each have much more finely structured basins of attraction, but none of the basin boundaries appear to be fractal. The basins of attraction for the chaotic and small-amplitude limit cycle dynamics apparently reside within each other. We briefly discuss the implications of these findings in the context of theoretical attempts to understand the dynamics of brain function and behaviour.     

Multiple attractors and periodic transients in synchronized nonlinear circuits

I study a pair of synchronized nonlinear circuits which may be periodic or chaotic. The circuits are synchronized by a one-way driving signal from the drive circuit to the response circuit. Because the nonlinearities are symmetric about zero, the drive circuit has two periodic attractors. When the value of a bifurcation parameter is above a certain threshold, the response circuit also has two periodic attractors, one in-sync with the drive and one out-of-sync. Below the threshold, the drive circuit still has two attractors but the response circuit has only one attractor, the in-sync attractor. If the response circuit is started in the basin of attraction of the former out-of-sync attractor, a long periodic transient (many cycles long) is seen.     

Synchronization of time-continuous chaotic oscillators

Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed, respectively, in terms of the subcritical, supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled R?ssler oscillators.     

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.