Dynamical Behavior of Coupled Chaotic Oscillators and its Control

作为耦合非线性振子模型，数值研究了由J格点组成的周期耦合映象格子（CML）。在该系统中，选择Logistic映射作为局部动力学，并将该映射的非线性参数从其他格点的2降为每M个格点的B。这些晶格点以耦合常数ε扩散耦合。如果J大到足以满足条件J <greater than or equal>5 M，则（B，B）平面上的相图除了少量的依赖于J外，对J的依赖性很弱。在相图中，我们发现了几个区域，称为区域A，B和C，在这些区域中观察到了模式选择。虽然对于小的λ，区域A从B = 0延伸到B = 2，这对应于均匀CML的情况，但是对于较大的λ，区域B和C仅延伸到小于2的某个B。在这些区域之外，我们通常观察到时空混沌。因此，当M不是很大时，如果B足够小，周期性的引入可以抑制区域B和C内的CML的模式选择的时空混沌。还发现，周期CML的解从随机初始状态到达模式选择的时间步长Nt如果B充分小于2，则在区域B和C中，以及在区域A中代数地随M大致指数地增加。因此，当选取较大的M时，虽然可以通过对均匀CML进行较小的修改来实现对时空混沌的抑制，但从随机初始状态达到这种抑制所需的时间步长要大得多。此外，作为非线性扩展系统的一个典型例子，考察了法拉第波斑图的形成。

As a model of coupled nonlinear oscillators, a periodic coupled map lattice (CML) composed of J lattice points are numerically examined. In this system, the logistic map is chosen as the local dynamics, and the nonlinearity parameter of this map is reduced to b for every Mth lattice points from 2 for other lattice points. These lattice points are diffusively coupled with the coupling constant epsilon. If J is large enough to satisfy the conditions J <greater than or equal> 5M the phase diagram on the (b, epsilon) plane depends on J only weakly except for a smalIn the phase diagrams, we find a few regions, called Regions A, B and C, in which the pattern selection is observed. Although Region A, found for small epsilon, extends from b = 0 to b = 2 which corresponds to the case of a homogeneous CML, Regions B and C, found for larger epsilon, extend only up to a certain b which is smaller than 2. Outside these regions, we usually observe the spatiotemporal chaos. Therefore, when M is not so large, the introduction of the periodicity can suppress the spatiotemporal chaos to the pattern selection for epsilon within Regions B and C if b is sufficiently small.It is also found that the transient length Nt, the time step at which a solution of the periodic CML reaches the pattern selection from a random initial state, increases with M roughly exponentially in Regions B and C and algebraically in Region A if b is sufficiently smaller than 2. Therefore, although the suppression of the spatiotemporal chaos is realized with smaller modification of the homogeneous CML if we choose larger M, the time step required for the attainment of this suppression from a random initial state becomes much larger.Furthermore, the pattern formation in Faraday waves is examined as a typical example of nonlinear extended system.