Spatially Periodic Orbits in Coupled Sine Circle Maps

We study spatially periodic orbits for a coupled map lattice of sine circle maps with nearest neighbour coupling and periodic boundary conditions. The stability analysis for an arbitrary spatial period k is carried out in terms of the independent variables of the problem and the stability matrix is reduced to a neat block diagonal form. For a lattice of size kN, we show that the largest eigenvalue for the stability matrix of size $kN \times kN$ is the same as that for the basic spatial period k matrix of size $k \times k$. Thus the analysis for a kN lattice case can be reduced to that for a k lattice case. We illustrate this explicitly for a spatial period two case. Our formalism is general and can be extended to any coupled map lattice. We also obtain the stability regions of solutions which have the same spatial and temporal period numerically. Our analysis shows that such regions form a set of Arnold tongues in the $\Omega-\epsilon-K$ space. The tongues corresponding to higher spatial periods are contained within the tongues seen in the temporally periodic spatial period one or synchronised case. We find an interesting new bifurcation wherein the the spatially synchronised and temporal period one solution undergoes a bifurcation to a spatio-temporal period two travelling wave solution. The edges of the stability interval of this solution are analytically obtained.