Relationships between the Limit of Predictability and Initial Error in the Uncoupled and Coupled Lorenz Models

In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems:the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model-in which the slow dynamics and the fast dynamics interact with each other-there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.

Incremental 4D-VAR assimilation scheme based on Lorenz model

Four-dimensional variational(4D-VAR) data assimilation method is a perfect data assimilation solution in theory, but the computational issue is quite difficult in operational implementation.The incremental 4D-VAR assimilation scheme is set up in order to reduce the computational cost. It is shown that the accuracy of the observations, the length of the assimilation window and the choice of the first guess have an important influence on the assimilation outcome through the contrast experiment. Compared with the standard 4D-VAR assimilation scheme, the incremental 4D-VAR assimilation scheme shows its advantage in the computation speed through an assimilation experiment.

Theoretical Basis and Application of an Analogue-Dynamical Model in the Lorenz System

The theoretical basis and application of an analogue-dynamical model(ADM) in the Lorenz system is studied.The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective.Primary analyses show that under the condition of appending disturbances in model parameters,the model errors of ADM are much smaller than those of the pure dynamical model(PDM).The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments.The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM,but when model errors are considerably small,the latter will be superior to the former.To overcome such a problem,the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.

Optimal Control and Bifurcation Issues for Lorenz-Rössler Model

Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-Rössler model is one of the interesting models because of the idea of consolidation of the two models: Lorenz and össler. This paper discusses the Lorenz-Rössler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-Rössler model is discussed and it uses the Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical examples and solutions for bifurcation cases and the optimal controlled system are carried out and shown graphically to show the effectiveness of the used procedure.

非线性Lorenz-Like系统的定性分析及控制

利用定性分析和优化控制方法,实现了Lorenz-Like模型的Hopf分支和优化控制,得到了模型的动力学演化机理和分支条件及变量同步控制器和误差系统的全局渐近稳定条件.

Lorenz系统混沌解序列可预报性的统计检验

利用 Lorenz简化热对流模式产生了吸引子区域内的混沌解序列 ,对截取的序列进行了平稳性和正态性检验 .按照 3种情形分别选取样本 ,并应用统计预报中的 ARMA模型、多元线性回归模型、多项式回归模型和均值生成函数模型等作出预报 .比较分析表明 :所选取方程组产生的混沌解序列 ,呈现非周期、非平稳、非正态特性等极不规则的分布 ,导致几种统计模型对于两个不稳定平衡状态间的不确定的突变情形基本失去了预报能力 ,系统行为几乎无法预测 ,其根本原因在于系统的混沌特性 .但在某一不稳定平衡状态内 ,序列段呈现振幅不断放大的准周期振荡 ,具有一定的规律性 ,几种统计模型预报的效果较好 ,说明在应用统计方法进行预报的前提下 ,系统行为存在着局部时段的有限的可预报性 .

大脑协同学、Lorenz模型和认知科学

从认知科学及其脑科学的前沿 ,探讨了大脑神经网络巨系统的功能特点 .知觉及思维的产生正是大脑复杂的神经网络巨系统中各子系统的协同综合作用的结果 .基于这一机制和特点 ,初步提出了大脑协同学的一些基本原理和方法 .由协同学的基本方程 ,我们定量导出Lorenz方程和Lorenz模型 .其中的两翼对应于大脑的两个半球 ,两翼之间的跳来跳去形象地描述思维 .由此说明生命在于混沌中的协同 .大脑协同学具有三个层次 :大脑的结构、活动和思维 ,及进一步的心身协同学 .在此 ,我们尝试以一种新的理论模型对认知科学及脑科学进行某些探讨 .

变分同化方法在Lorenz系统中的简单应用研究

利用Lorenz模式作变分同化数值试验 ,通过对一个简单系统的讨论 ,介绍四维变分同化方法。对初值敏感性和观测点的个数及观测值作了对比试验 ,发现随着模式对初值敏感性的增加 ,同化效果会越来越差 ;观测点越少 ,观测值误差越大 ,这些都会影响同化效果 ,甚至导致同化失败。

Lorenz系统的距平模式和标准化距平模式

本文对Lorenz系统作时间积分获得模式的一个现实,利用此现实求得模式变量A的均值和方差的估计值A,σ_A,分别作线性变换代和Lorenz系统中,则得关于A'和A''的方程称为距平模式和标准化距平模式,两个模式的数值试验结果表明:即使对于Lorenz系统这样一个高度理想化的非线性系统,它的距平和标准化距平模式的统计特征和原模型接近,且标准化距平模式的结果更接近于原模型,这为我们在长期数值天气预报中使用距平模式和标准化距平模式提供了一个例证。

Lorenz混沌序列的改进

对Lorenz混沌序列进行了较深入的时频分析,进而建立数学模型,实现了对其固有时频缺陷的全面改进,并利用统计检验进行了随机性验证。分析和仿真表明,改进后的Lorenz混沌序列分布均匀,并有较理想的相关特性;其相空间轨迹变得极不规则,更难于相空间重构;对高频成分的放大,使得该序列有着类白噪声的频谱特性。

修正的Lorenz模型的Lyapunov指数

采用非线性动力学的分析方法，求出了修正的Ｌｏｒｅｎｚ模型的Ｌｙａｐｕｎｏｖ指数，确定出该体系存在着混沌吸引子。

广义Lorenz系统的模糊建模和同步

广义Lorenz系统包括经典Lorenz系统和无穷多个结构相似但不拓扑等价的混沌系统。针对广义Lorenz规范式(GLCF)结构,基于T-S模型进行了系统的模糊重构;利用Lyapunov稳定性理论和反馈同步的思想实现了重构系统的同步,并推导了误差系统以衰减率α实现全局渐近稳定的充分条件。统一混沌系统和扩展Liu系统均属于典型的广义Lorenz系统。利用LMI方法对这两个混沌系统所进行的仿真实验结果表明,该模糊同步方法对于满足GLCF的所有混沌系统均适用。

基于Lorenz规范的空间-波数域大地电磁三维正演

大地电磁勘探方法由于其成本低、施工简单、探测深度广等优点,广泛应用于矿产资源普查、油气勘探和深部构造研究等领域.如何提高大规模三维大地电磁数值模拟的精度和效率一直是研究热点.本文基于空间-波数域方法,实现了基于Lorenz规范的空间-波数域三维大地电磁数值模拟.基于二次场计算原理,引入Lorenz规范,将Maxwell方程组转化为关于二次场矢量位的亥姆霍兹方程;利用水平方向二维傅里叶变换,将空间域三维偏微分方程转换为多个波数下相互独立的常微分方程,方程采用二次插值有限单元法计算,得到定带宽线性方程组,方程计算量小、并行性好,采用追赶法求解,提高了算法效率;引入压缩算子,用迭代法逐次逼近真实解.充分利用了空间-波数域方法数值精度高、内存需求少、效率高的特点.设计棱柱体模型验证了算法的正确性、分析了算法的收敛性,说明算法对不同频率、不同电导率对比度模型均具有很好的适应性.利用Dublin(DTM1)模型进行三维大地电磁数值模拟,结果表明:在满足精度要求的前提下,空间-波数域算法比空间域算法占用内存少、耗时短;相比基于Coulomb规范的空间-波数域算法,基于Lorenz规范的空间-波数域方法耗时更短、占用内存更少,效率提高至少3倍以上,体现了新方法的优势.

单模激光Lorenz系统与Duffing系统缩阶反同步

对于2阶Duffing系统与3阶单模激光Lorenz系统,用主动控制方法设计了一种同步控制器,实现二者之间的缩阶反同步。依据罗斯-霍维兹判据,给出误差系统在原点处渐进稳定的条件。数值试验证实:在控制器作用下,2阶Duffing系统的状态变量与3阶单模激光Lorenz投影子系统的状态变量振动的幅值相同但方向相反,且在随机噪声干扰下反同步仍能很好地实现。这表明所设计的控制器对于不同阶的Duffing系统与Lorenz系统的缩阶反同步的实现是可行的、有效的,而且具有很好的鲁棒性。

基于Lorenz模型的集合预报与单一预报的比较研究

根据非线性局部Lyapunov向量方法和增长模繁殖方法,选取Lorenz63模型和Lorenz96模型的不同状态为例,对集合预报与单一预报的预报技巧开展了对比研究.结果表明:与单一预报比较,集合预报的均方根误差和型异常相关有明显改善,随预报时间推移,改善效果越显著,且集合平均优于单一预报的实验个例数逐渐增多.就概率分布(f)而言,单一预报状态的f与真实状态基本一致,不随时间变化;而集合平均预报状态的f则随时间呈现出值域变窄、峰值变大的特点.表明随预报时间的延长,单一预报状态为混沌吸引子上的随机状态,而集合平均预报状态为吸引子子集上的随机状态,这可能是集合平均误差小于单一预报的原因.

控制Lorenz系统的新方法

讨论Lorenz混沌系统的控制问题.首先将Lorenz系统用模糊T-S模型来表示,然后引入脉冲控制律,利用脉冲微分系统理论进行分析,得到了Lorenz系统的脉冲控制准则.根据该准则所设计的脉冲控制器,可以驱动受控Lorenz系统的轨线以指数衰减率收敛于平衡点.

旋转的Rayleigh-Bénard问题的Lorenz模型及数值模拟

利用截谱方法,将旋转的Rayleigh-Bénard问题转化为四维的Lorenz模型,估计出此Lorenz模型的参数取值范围为τ∈[0,2],r∈(0,∞),b∈[0,8/3]。比较了四维Lorenz方程与经典的三维的Lorenz方程(无旋转)的不同之处,在此基础上进行了数值模拟,结果表明旋转可以增加系统稳定性。

Neutrality Criteria for Stability Analysis of Dynamical Systems through Lorentz and Rossler Model Problems

Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.

Atmospheric Convection Model Based Digital Confidentiality Scheme

Nonlinear dynamics is a fascinating area that is intensely affecting a wide range of different disciplines of science and technology globally.The combination of different innovative topics of information security and high-speed computing has added new visions into the behavior of complex nonlinear dynamical systems which uncovered amazing results even in the least difficult nonlinearmodels.The generation of complex actions froma very simple dynamical method has a strong relation with information security.The protection of digital content is one of the inescapable concerns of the digitally advanced world.Today,information plays an important role in everyday life and affects the surroundings rapidly.These digital contents consist of text,images,audio,and videos,respectively.Due to the vast usage of digital images in the number of social and web applications,its security is one of the biggest issues.In this work,we have offered an innovative image encryption technique based on present criteria of confusion and diffusion.The designed scheme comprises two major nonlinear dynamical systems.We have employed discrete fractional chaotic iterative maps to add confusion capability in our suggested algorithm and continuous chaotic Lorenz system.We have verified our offered scheme by using statistical analysis.The investigations under the statistical tests suggested that our proposed technique is quite reasonable for the security of digital data.