Chains of N unidirectionally coupled nonlinear first-order equations are considered, where the value of the last element is determined through the first element of the chain. The aim of this work is to investigate the local – in the neighborhood of the zero equilibrium state – dynamics of this system. Critical cases in the problem of equilibrium state stability are identified, and normal forms determining the local behavior of solutions are constructed. A detailed analysis is performed in the simplest cases, where N = 2 and N = 3. The most interesting part of the research concerns the case where the value of N is sufficiently large. It is shown that the critical cases then have infinite dimension.

Methods. The standard research scheme, based on the use of the method of local invariant manifolds and the method of normal forms, turns out to be inapplicable. A special method of infinite-dimensional normalization developed by the author is used. The main results consist in the construction of so-called quasi-normal forms – analogs of normal forms for the infinite-dimensional case. It is important to emphasize that even for sufficiently large values of the number of chain elements N, the quasi-normal forms determining the dynamics of the original system significantly depend on variations in the value of N. Note that for certain values of the system coefficients, its dynamics can be quite complex.