一维耦合映象格子中参数噪声影响的研究
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摘要
系统复杂的终态行为具有普遍性，其中混沌行为很典型。混沌开创了复杂性演化的细致研究，破除了确定性演化是可预测的传统观念。任何现实系统都会受到噪声的扰动，人们自然要考虑到在噪声扰动下的系统动力学性态。现在人们已经逐渐认识到混沌与噪声之间的一些类似之处，这更激发人们对混沌系统在噪声环境下的性态表现的研究兴趣，特别是具有很强现实意义的时空混沌系统在噪声影响下的表现更加受到了特别关注。本文的工作是以一维耦合映象格子作为研究模型，这是一个被广泛加以研究的时空混沌系统的简化模型，重点考虑了局部动力学参数受到扰动的情况下系统的性态表现，并将之主要应用于目前的混沌同步中，取得了一定阶段性的结果。本文的主要内容如下：
1、本文首先概述了目前混沌学研究的目前进展，指出对混沌的深入的研究正在促进复杂性科学的形成，并给出了一般性进行混沌研究的基本工具。对于目前混沌应用研究的重要研究领域—混沌控制与同步给予了扼要的介绍，并对目前的噪声在混沌控制与同步的应用给予了较为具体的评述。
2、考虑一维对称耦合映象格子系统中参数噪声的存在，并给出了具体可能的噪声存在形式，并给予了物理解释。对噪声系统的最大Lyapunov指数的数值计算方法进行了研究。这是判断是否是混沌态
广西大学硕士研究生学位论文
中文摘要
的重要判据。分析认为只要噪声系统的MLE小于零，就存在着应用嗓
声将两个混沌系统驱动同步的可能。
3、应用了最大幼.pun0Pv指数具体研究了少体和一般的CM工s
在噪声影响下的系统性态表现，发现了对噪声响应敏感的系统参数区
域，并重点研究了系统在噪声影响的一些性态表现，发现存在对噪声
的响应是非平庸的，较为复杂的敏感参数区间。局域噪声基本很难能
导致系统整体性态的改变，系统具有自发的抵抗噪声的能力。
4、对于应用参数噪声驱动O旧JS同步作了研究，一定程度上肯定
了作者对噪声驱动同步的预测，但是具体表现是非平庸的，并非简单
的由最大切.punov指数唯一的决定，而且还与噪声的负向随机波动
的幅度有关，负向随机波动越大，则越有可能获得同步，在这个过程
中，同步偏差逐渐显示出结构，这反映系统趋向于越来越有序。同时
考察了徽小噪声对系统的性态的迁移控制作用。
5、初步探讨了同步的具体过程，发现全局随机同步过程是一种渐
进的过程，同步过程首先出现局域格点同步或局域格点集团(cl uster)
同步。发现全局非同步下的局域同步，但是在一定的精度要求内。同
时还发现存在噪声影响下的更弱的一种同步，局域随机相同步，即只
有对应的格点之间的相位保持一致，但是状态值不相同。同时对局域
格点的稳定性进行了初步讨论，利用本文定义的局域稳定指数，可以
有效地预侧出噪声环境下局域格点的演化性态。
本文获得了一些关于系统性态是如何响应噪声的影响的有价值
的结果，并且应用噪声对CML召的控制与同步进行了探究碍到了阶
广西大学硕士研究生学位论文
中文摘要
段性的成果。关于噪声驱动同步的内在机制以及局域同步和局域相同
步的特性量的刻画和预侧尚在进一步深入的研究中。
本文的工作受助于国家自然科学基金。(基金批准号:10147201)
The complicated long-term state is very common in the nature systems of which chaos is very typical. Chaos initiates meticulous study of the complicated evolutive behavior and explodes the traditional concept of being forcasted to the determinate evolvement. As any system in the reality exists in the noise surroundings, we should take the affect of noise very seriously if the noise is inevitable and continual. Now it has been already realized that there is some similarity between noise and chaos . People has the great interest to the noise affect on the chaos systems, especially the noise affect on the Spatiotemporal chaos.
Taken the one-dimensional coupled map lattices(CMLs) which is introduced as a simple model with the essential features of spatiotemporal systems as our study model, the affect of parameter noise on one-dimensional coupled map lattices and stochastic synchronization has been studied in this paper. The main results are listed as follows:
1 In the first section we summarize the progress in the chaos study and point out that study of chaos deeply is promoting the formation of the complex science. The basic tools for the study of chaos and the important
field of chaos application-chaos control and synchronization are
introduced compactly. At the same time , we emphases expatiating the application of the noise affect on chaos control and synchronization.
2 The parameter noise existing in the one-dimensional coupled map lattices is considered and different kinds of noise are gived. The numerical computation of The Maximum Lyapunov Exponent (MLE) has been studied which decided whether the system exists in the chaos state. We analyze that if the MLE of random CMLs is negative, there exits the possibility of driving the two CMLs to synchronize.
3 we take the CMLs of few-body and many-body as our study model. The change of state of CMLs under the noise affect by the MLE is studied and we found there exit systems parameter fields very sensitive to noise which show complicated behavior. Local noise is very difficult to change the state of whole system which has the ability of standing against the noise spontaneously.
4 we study the synchronization of CMLs under the noise driving and support our forecast to some extent which also express the complicated behavior. The synchronization is not only decided by the MLE ,but also decided by the range of the random negative fluctuation . The possibility for synchronization of CMLs will increase with the enlarge the negative random fluctuation range. During this course, synchronization windage will show the structure along the time steps, it means systems tend to order little by little. We also find the little noise can induce the CMLs to
periodic state or chaos.
5 we study the synchronization process and find the global synchronization is a gradual process that the local synchronization or cluster synchronization appear firstly. In the some limited precision, we can find the local synchronization in the loss of synchronization of whole CMLs systems. At the same time ,it exits a weaker local synchronization-local phase synchronization which means the phase of corresponding lattices is locked, but state variable is different. Further more ,the stability of the local lattice is probed into and the long-term evolutive behavior of the local lattice in the noise surroundings can be forecast effectively through the local characteristic exponent defined by this article.
This article obtained some results of the CMLs how to response the parameter noise .At same time, the synchronization under the parameter noise driving can be obtained through this article study . The inherent mechanism about the synchronization under the parameter noise driving and local synchronization phase synchronization will need to farther study deeply.
This Project is supported by the National Natural Science Foundation (Grant No.10147201)
1、本文首先概述了目前混沌学研究的目前进展，指出对混沌的深入的研究正在促进复杂性科学的形成，并给出了一般性进行混沌研究的基本工具。对于目前混沌应用研究的重要研究领域—混沌控制与同步给予了扼要的介绍，并对目前的噪声在混沌控制与同步的应用给予了较为具体的评述。
2、考虑一维对称耦合映象格子系统中参数噪声的存在，并给出了具体可能的噪声存在形式，并给予了物理解释。对噪声系统的最大Lyapunov指数的数值计算方法进行了研究。这是判断是否是混沌态
广西大学硕士研究生学位论文
中文摘要
的重要判据。分析认为只要噪声系统的MLE小于零，就存在着应用嗓
声将两个混沌系统驱动同步的可能。
3、应用了最大幼.pun0Pv指数具体研究了少体和一般的CM工s
在噪声影响下的系统性态表现，发现了对噪声响应敏感的系统参数区
域，并重点研究了系统在噪声影响的一些性态表现，发现存在对噪声
的响应是非平庸的，较为复杂的敏感参数区间。局域噪声基本很难能
导致系统整体性态的改变，系统具有自发的抵抗噪声的能力。
4、对于应用参数噪声驱动O旧JS同步作了研究，一定程度上肯定
了作者对噪声驱动同步的预测，但是具体表现是非平庸的，并非简单
的由最大切.punov指数唯一的决定，而且还与噪声的负向随机波动
的幅度有关，负向随机波动越大，则越有可能获得同步，在这个过程
中，同步偏差逐渐显示出结构，这反映系统趋向于越来越有序。同时
考察了徽小噪声对系统的性态的迁移控制作用。
5、初步探讨了同步的具体过程，发现全局随机同步过程是一种渐
进的过程，同步过程首先出现局域格点同步或局域格点集团(cl uster)
同步。发现全局非同步下的局域同步，但是在一定的精度要求内。同
时还发现存在噪声影响下的更弱的一种同步，局域随机相同步，即只
有对应的格点之间的相位保持一致，但是状态值不相同。同时对局域
格点的稳定性进行了初步讨论，利用本文定义的局域稳定指数，可以
有效地预侧出噪声环境下局域格点的演化性态。
本文获得了一些关于系统性态是如何响应噪声的影响的有价值
的结果，并且应用噪声对CML召的控制与同步进行了探究碍到了阶
广西大学硕士研究生学位论文
中文摘要
段性的成果。关于噪声驱动同步的内在机制以及局域同步和局域相同
步的特性量的刻画和预侧尚在进一步深入的研究中。
本文的工作受助于国家自然科学基金。(基金批准号:10147201)
The complicated long-term state is very common in the nature systems of which chaos is very typical. Chaos initiates meticulous study of the complicated evolutive behavior and explodes the traditional concept of being forcasted to the determinate evolvement. As any system in the reality exists in the noise surroundings, we should take the affect of noise very seriously if the noise is inevitable and continual. Now it has been already realized that there is some similarity between noise and chaos . People has the great interest to the noise affect on the chaos systems, especially the noise affect on the Spatiotemporal chaos.
Taken the one-dimensional coupled map lattices(CMLs) which is introduced as a simple model with the essential features of spatiotemporal systems as our study model, the affect of parameter noise on one-dimensional coupled map lattices and stochastic synchronization has been studied in this paper. The main results are listed as follows:
1 In the first section we summarize the progress in the chaos study and point out that study of chaos deeply is promoting the formation of the complex science. The basic tools for the study of chaos and the important
field of chaos application-chaos control and synchronization are
introduced compactly. At the same time , we emphases expatiating the application of the noise affect on chaos control and synchronization.
2 The parameter noise existing in the one-dimensional coupled map lattices is considered and different kinds of noise are gived. The numerical computation of The Maximum Lyapunov Exponent (MLE) has been studied which decided whether the system exists in the chaos state. We analyze that if the MLE of random CMLs is negative, there exits the possibility of driving the two CMLs to synchronize.
3 we take the CMLs of few-body and many-body as our study model. The change of state of CMLs under the noise affect by the MLE is studied and we found there exit systems parameter fields very sensitive to noise which show complicated behavior. Local noise is very difficult to change the state of whole system which has the ability of standing against the noise spontaneously.
4 we study the synchronization of CMLs under the noise driving and support our forecast to some extent which also express the complicated behavior. The synchronization is not only decided by the MLE ,but also decided by the range of the random negative fluctuation . The possibility for synchronization of CMLs will increase with the enlarge the negative random fluctuation range. During this course, synchronization windage will show the structure along the time steps, it means systems tend to order little by little. We also find the little noise can induce the CMLs to
periodic state or chaos.
5 we study the synchronization process and find the global synchronization is a gradual process that the local synchronization or cluster synchronization appear firstly. In the some limited precision, we can find the local synchronization in the loss of synchronization of whole CMLs systems. At the same time ,it exits a weaker local synchronization-local phase synchronization which means the phase of corresponding lattices is locked, but state variable is different. Further more ,the stability of the local lattice is probed into and the long-term evolutive behavior of the local lattice in the noise surroundings can be forecast effectively through the local characteristic exponent defined by this article.
This article obtained some results of the CMLs how to response the parameter noise .At same time, the synchronization under the parameter noise driving can be obtained through this article study . The inherent mechanism about the synchronization under the parameter noise driving and local synchronization phase synchronization will need to farther study deeply.
This Project is supported by the National Natural Science Foundation (Grant No.10147201)
引文
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