局部耦合不连续映象格子的集体动力学研究
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摘要
本文研究了局部耦合既不连续又不可逆映象格子系统的集体动力学特征。分别选择映象处于周期5,周期5和11带混沌共存,周期5以及6带混沌等四种状态,计算、并分析了耦合系统同步的同步序参数Rn和最大李雅普诺夫指数λmax。并且通过空间振幅变化图和时空行为发展图观察系统时空动力学过程的细节。
研究结果发现,当单映象出于周期5状态时,系统在不同耦合强度下表现出三种不同的动力学,即,冻结化随机图案模式,同步和平移,图样竞争阵发混沌模式；当单映象处于周期5和11带混沌共存状态时,系统在较小的耦合强度下就可以达到同步；当单映象处于6带混沌状态的,出现了相同步现象。同时,仔细地分析了产生上述现象的原因,描述了其特征。
本工作的最重要结果就是,发现了在系统同步前的一类具有特殊结构的前奏动力学,以及与此前奏动力学类似的时间周期动力学。对于第一类现象的分析发现,空间被分割为分别同步于单映象的两个相邻周期轨道点的两个同步区间。随着时间的增加增加一个扩张、另一个收缩直至出现整个空间的完全同步。进一步的研究又发现,上述过程实际上伴随着左行和右行类孤波的对产生,类孤波对将空间分割为两个子区域,类孤波的移动造成一个子区间扩张、另一个子区间收缩。当收缩区消失时,出现两类孤波碰撞,这种碰撞造成其对湮灭,并同时产生一个“耗散的呼吸子”,该呼吸子的衰变直至死亡引发完全的同步。对于第二类现象,系统随着时间的演化没有出现同步现象,而同步序参量的时间序列呈现一种周期性的变化。在每个周期内,出现一个与上述前奏动力学结构相同的子过程,和一个恢复子过程。前一子过程中,系统空间中的两个子区间的变化与前奏结构的变化方式相同。该子过程结束于收缩集团的消失,此时两集团的边界融合为一个小的空间区域。此后,恢复子过程开始,融合区中格点上的动力学变量经历一个调整过程直到其中的一个格点变为新的扩张集团的种子。随后,又一个循环开始,周而复始。该现象,也可以用类孤波和呼吸子的概念分析。依据上述分析的结果,通过引入指向相的概念,得到了这种特殊前奏动力学和周期动力学中序参量的近似解析解,与数值解符合的很好。
The collective dynamics is of the locally coupled map lattices consisting of the both discontinuous and non-invertible maps. in the same state of period-5, period-5 coexisting with band-11 chaos, period-5 and band-6 chaos. The order parameters for synchronization and the largest Lyapunov exponents are calculated and analyzed when the states of the single map are in the period-5, the coexistence of period-5 and band-11 chaos, another period-5, and the band-6 chaos states, respectively. The details of the spatiotemporal dynamics of the system are observed through the space-amplitude plots and space-time diagrams.
The result reveals that there are three kinds of dynamics when the single map is in its period-5 state. They are the frozen random patterns, the synchronization and the fully developed turbulence. The synchronization appears at the smaller coupling strength when the state of the single map is in the coexistence state of the period-5 and the band-11 chaos. The phase synchronization can be observed as the single map is in the state of band-6 chaos. The mechanisms for these phenomena are analyzed and their characteristics are described.
The most important results in this work are the discovery of a peculiar prelude dynamics before synchronization and a periodic dynamics with a similar structure to the prelude dynamics. The analysis on the first type of the phenomena reveals that the spatial sites are decomposed into two synchronized clusters which synchronize to adjacent trajectories of the periodic orbit of the single map. With the time evolution, one of them expands but another one contracts until all the sites approach to the same synchronized state. The further study shows that this process is actually related to the pair-creation of a left-moving and a right-moving soliton-like waves These two soliton-like waves separate the space into two sub-clusters, and the moving of the two soliton-like waves causes the expanding of one cluster and the contracting of the other one. They collide with each other when the contracting cluster disappears, which results in the pair-elimination of the two soliton-like waves and produces a "dissipative breather" simultaneously. The breather decays, and its death induces the complete synchronization. For the second type of the phenomena, the system does not synchronize with time, but the order parameter varies periodically as the time increase. There appear a sub-process that has the same structure to the above-mentioned prelude dynamics and a recovery sub-process that pushes the system back to the beginning of the prelude dynamics in each cycle. In the former sub-process, the two sub-clusters in the system space evolve in the same way to the prelude structure. This sub-process ends up with the disappearing of the contracting cluster where the boudoirs of the two clusters merge into a small region in the space. Then the recovery process begins, and the dynamical variables in the merged region adjust themselves until a site in the middle of the region becomes the seed of a new expanding cluster. Right after this, a new cycle starts, and this process repeats till infinity. This phenomenon can also be explained by the concepts of soliton and breather. Based upon the analysis above, the analytical expressions of the order parameter in the peculiar prelude dynamics and the periodic dynamics are obtained through the introduction of the directory phase, which fits into the numerical results very well.
研究结果发现,当单映象出于周期5状态时,系统在不同耦合强度下表现出三种不同的动力学,即,冻结化随机图案模式,同步和平移,图样竞争阵发混沌模式；当单映象处于周期5和11带混沌共存状态时,系统在较小的耦合强度下就可以达到同步；当单映象处于6带混沌状态的,出现了相同步现象。同时,仔细地分析了产生上述现象的原因,描述了其特征。
本工作的最重要结果就是,发现了在系统同步前的一类具有特殊结构的前奏动力学,以及与此前奏动力学类似的时间周期动力学。对于第一类现象的分析发现,空间被分割为分别同步于单映象的两个相邻周期轨道点的两个同步区间。随着时间的增加增加一个扩张、另一个收缩直至出现整个空间的完全同步。进一步的研究又发现,上述过程实际上伴随着左行和右行类孤波的对产生,类孤波对将空间分割为两个子区域,类孤波的移动造成一个子区间扩张、另一个子区间收缩。当收缩区消失时,出现两类孤波碰撞,这种碰撞造成其对湮灭,并同时产生一个“耗散的呼吸子”,该呼吸子的衰变直至死亡引发完全的同步。对于第二类现象,系统随着时间的演化没有出现同步现象,而同步序参量的时间序列呈现一种周期性的变化。在每个周期内,出现一个与上述前奏动力学结构相同的子过程,和一个恢复子过程。前一子过程中,系统空间中的两个子区间的变化与前奏结构的变化方式相同。该子过程结束于收缩集团的消失,此时两集团的边界融合为一个小的空间区域。此后,恢复子过程开始,融合区中格点上的动力学变量经历一个调整过程直到其中的一个格点变为新的扩张集团的种子。随后,又一个循环开始,周而复始。该现象,也可以用类孤波和呼吸子的概念分析。依据上述分析的结果,通过引入指向相的概念,得到了这种特殊前奏动力学和周期动力学中序参量的近似解析解,与数值解符合的很好。
The collective dynamics is of the locally coupled map lattices consisting of the both discontinuous and non-invertible maps. in the same state of period-5, period-5 coexisting with band-11 chaos, period-5 and band-6 chaos. The order parameters for synchronization and the largest Lyapunov exponents are calculated and analyzed when the states of the single map are in the period-5, the coexistence of period-5 and band-11 chaos, another period-5, and the band-6 chaos states, respectively. The details of the spatiotemporal dynamics of the system are observed through the space-amplitude plots and space-time diagrams.
The result reveals that there are three kinds of dynamics when the single map is in its period-5 state. They are the frozen random patterns, the synchronization and the fully developed turbulence. The synchronization appears at the smaller coupling strength when the state of the single map is in the coexistence state of the period-5 and the band-11 chaos. The phase synchronization can be observed as the single map is in the state of band-6 chaos. The mechanisms for these phenomena are analyzed and their characteristics are described.
The most important results in this work are the discovery of a peculiar prelude dynamics before synchronization and a periodic dynamics with a similar structure to the prelude dynamics. The analysis on the first type of the phenomena reveals that the spatial sites are decomposed into two synchronized clusters which synchronize to adjacent trajectories of the periodic orbit of the single map. With the time evolution, one of them expands but another one contracts until all the sites approach to the same synchronized state. The further study shows that this process is actually related to the pair-creation of a left-moving and a right-moving soliton-like waves These two soliton-like waves separate the space into two sub-clusters, and the moving of the two soliton-like waves causes the expanding of one cluster and the contracting of the other one. They collide with each other when the contracting cluster disappears, which results in the pair-elimination of the two soliton-like waves and produces a "dissipative breather" simultaneously. The breather decays, and its death induces the complete synchronization. For the second type of the phenomena, the system does not synchronize with time, but the order parameter varies periodically as the time increase. There appear a sub-process that has the same structure to the above-mentioned prelude dynamics and a recovery sub-process that pushes the system back to the beginning of the prelude dynamics in each cycle. In the former sub-process, the two sub-clusters in the system space evolve in the same way to the prelude structure. This sub-process ends up with the disappearing of the contracting cluster where the boudoirs of the two clusters merge into a small region in the space. Then the recovery process begins, and the dynamical variables in the merged region adjust themselves until a site in the middle of the region becomes the seed of a new expanding cluster. Right after this, a new cycle starts, and this process repeats till infinity. This phenomenon can also be explained by the concepts of soliton and breather. Based upon the analysis above, the analytical expressions of the order parameter in the peculiar prelude dynamics and the periodic dynamics are obtained through the introduction of the directory phase, which fits into the numerical results very well.
引文
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[9]齐洁,汪定伟.双头垄断市场中广告模型的混沌动态特性[J].控制理论与应用,2005,22(2)：273～276.
[10]T. Wang, K. J. Wang, N. Jia. Chaos Control and Associative Memory of a Time-Delay Globally Coupled Neural Network Using Symmetric Map[J]. Neurocomputing,2011,74:1673-1680.
[11]格雷克.混沌：开创新科学[M].北京：高等教育出版社,2004：150～153.
[12]S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, C. S. Zhou. The Synchronization of Chaotic Systems[J]. Physics Report,2002,366:1-101.
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