Coupled map lattices with complex order parameter

We introduce and study coupled map lattices with complex state variable. The dynamical regimes of the finite-length systems are classified naturally in terms of a topological invariant – the overall phase shift accumulated along the whole length at fixed time. A stability analysis of the spatially uniform states is presented, and the results of numerical simulations of the spatio-temporal dynamics are discussed. We demonstrate that fast amplitude evolution, including regular and chaotic spatio-temporal behavior, takes place on the background of a slower phase evolution. For large values of the topological invariant the phase dynamics may give rise to an instability, which in some cases results in a jump of the system to another value of the invariant. We also consider the formation of long-lived “bubbles”, i.e., local domains of complicated dynamics in the spatial regions of locally reduced phase gradient. Our coupled map lattice model and its generalizations may be useful for understanding the dynamics in a larger range of parameters for such nonlinear dissipative media, which allow small-amplitude description in terms of the complex Ginzburg–Landau equation, as well as for time-delay feedback systems with nonzero central frequency of the generated signal.