Spatiotemporal stability of synchronized coupled map lattice states

• A spatiotemporal orbit Jacobian is computed for coupled map lattices in the first Bruillouin zone. • The linear stability of synchronized states under periodic and aperiodic perturbations is studied. • Short periodic orbits exhibit a nontrivial dependence of the stability exponent on the lattice coupling. In the realm of spatiotemporal chaos, unstable periodic orbits play a major role in understanding the dynamics. Their stability changes and bifurcations in general are thus of central interest. Here, coupled map lattice discretizations of nonlinear partial differential equations, exhibiting a variety of behaviors depending on the coupling strength, are considered. In particular, the linear stability analysis of synchronized states is performed by evaluating the Bravais lattice orbit Jacobian in its reciprocal space first Brillouin zone, with space and time treated on equal grounds. The eigenvalues of the orbit Jacobian operator, computed as functions of the coupling strength, tell us about the stability of the periodic orbit under a perturbation of a certain time- and space frequency. Moreover, the stability under aperiodic, that is, incoherent perturbations, is revealed by integrating the sum of the stability exponents over all space-time frequencies.