Stability of spatially homogeneous chaotic regimes in unidirectional chains

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Abstract
Necessary and sufficient stability conditions for spatially homogeneous stochastic regimes are considered in the chains of unidirectionally coupled one-dimensional maps. According to the form of a linearised system, two types of coupling between the elements of the chain are distinguished: dissipative and quasiinertial coupling. For the dissipative coupling, necessary and sufficient stability conditions are found for two different types of stability: homogeneous (when initial excitations are damped in all elements of the chain simultaneously) and weak stability (when the time of the perturbation damping depends significantly on the departure of the elements from the end of the chain). For the quasi-inertial coupling, necessary and sufficient conditions are found for time-periodic spatially homogeneous solutions. Sufficient stability conditions for stochastic regimes in the general case were not obtained. Numerical experiment revealed stable stochastic regimes in the parameter regions predicted by the theory. In chains with quasi-inertial coupling stable spatially homogeneous regimes were not observed.