Dynamic and stochastic effects of the interaction of individual localized structures and lattices of structures of a nonlinear field are considered within one-dimensional large-box Ginzburg-Landau and Swift-Hohenberg models. The criterion of soliton “survival” in competition with counterpropagating modes is derived analytically for a quasigradient case. The criterion of linear stability of a spatially homogeneous regime, that is similar to the Benjamin-Feir condition, is obtained for coupled Ginzburg-Landau equations. The relation between the correlation dimensions of the space series of counterpropagating modes and the dimension of the time series is investigated. It is shown that within a quasigradient model of two weakly coupled counterpropagating modes, the field can be represented as a superposition of two fixed stochastic spatial lattices which correspond to these modes and move through one another at constant velocity. The dimension of the time series at a given point in space is close to the sum of dimensions of the spatial distributions of counterpropagating modes. With the increase of coupling, the interaction of coupled modes “loosens” the equilibrium state corresponding to fixed lattices and the dimension of the time series grows.
Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs