Pradhan Gauri R, Chatterjee Nandini, Gupte Neelima
Department of Physics, University of Pune, Pune 411007, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Apr;65(4 Pt 2A):046227. doi: 10.1103/PhysRevE.65.046227. Epub 2002 Apr 10.
We study the organization of mode-locked intervals corresponding to the stable spatiotemporally periodic solutions in a lattice of diffusively coupled sine circle maps with periodic boundary conditions. Spatially periodic initial conditions settle down to spatiotemporally periodic solutions over large regions of the parameter space. In the case of synchronized solutions resulting from synchronized initial conditions, the mode-locked intervals have been seen to follow strict Farey ordering in the temporal periods. However, the nature of the organization of the mode-locked intervals corresponding to higher spatiotemporal periods is highly dependent on initial conditions and on system parameters. Farey ordering in the temporal periods is seen at low coupling for mode-locked intervals of all spatial periods. On the other hand, stable spatial period two solutions show an interesting reversal of Farey ordering at high values of coupling. Other spatially periodic solutions show a complete departure from Farey ordering at high coupling. We also examine the issue of completeness of the mode-locked intervals via a calculation of the fractal dimension of the complement of the mode-locked intervals as a function of the coupling epsilon and the nonlinearity parameter K. Our results are consistent with completeness over a range of values for these parameters. Spatiotemporally periodic solutions of the traveling wave type have their own organization in the parameter space. Novel bifurcations to other types of solutions are seen in the mode-locked intervals. We discuss various features of these bifurcations. We also define a set of new variables using which an analytic treatment of the bifurcations along the Omega=0 line is carried out.
我们研究了在具有周期边界条件的扩散耦合正弦圆映射晶格中，与稳定的时空周期解相对应的锁模区间的组织形式。空间周期初始条件在参数空间的大区域内收敛到时空周期解。在由同步初始条件产生的同步解的情况下，锁模区间在时间周期上遵循严格的法雷序列。然而，对应于更高时空周期的锁模区间的组织性质高度依赖于初始条件和系统参数。对于所有空间周期的锁模区间，在低耦合时在时间周期上可看到法雷序列。另一方面，稳定的空间周期二解在高耦合值时显示出法雷序列的有趣反转。其他空间周期解在高耦合时完全偏离法雷序列。我们还通过计算锁模区间补集的分形维数作为耦合ε和非线性参数K的函数，研究了锁模区间的完备性问题。我们的结果与这些参数在一定取值范围内的完备性一致。行波型的时空周期解在参数空间中有其自身的组织形式。在锁模区间中可看到向其他类型解的新型分岔。我们讨论了这些分岔的各种特征。我们还定义了一组新变量，利用它们对沿Ω = 0线的分岔进行解析处理。
