Mode locking of spatiotemporally periodic orbits in coupled sine circle map lattices.

Abstract

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.