Dynamics of Non-Periodic Chains with One-Sided and Two-Sided Couplings

This paper considers the question of local dynamics of the simplest non-periodic chains of nonlinear first-order equations with two-sided couplings. The main attention is paid to the study of chains with a large number N of elements. The critical cases in the problem of stability of the zero equilibrium state are identified. Questions about bifurcations of regular and irregular solutions are considered. Analogues of normal forms are constructed, the so-called quasinormal forms, which are special nonlinear equations of parabolic type. Their nonlocal dynamics determine the local structure of solutions to the original problem. Bifurcation problems for quasinormal forms are considered, and interestingly, the boundary conditions for them are not classical. The asymptotics of both regular and irregular solutions are constructed. The latter have the most complex structure. In particular, for negative values of the coupling parameter between elements, continual families of equilibrium states, cycles, and more complex structures can arise in the chain.