Nonperiodic Chains of Second-Order Equations with One-Sided Coupling

We consider the local dynamics (in a neighborhood of the equilibrium state) of nonperiodic chains of $N$ identical elements, each described by a second-order equation. It is assumed that the coupling between chain elements is one-sided. The main assumption is that the number $N$ of elements is sufficiently large, i. e., the small parameter is $N^{−1}$. Critical cases in the stability problem for the zero solution are identified, and it is shown that they have infinite dimension in the sense that infinitely many roots of the corresponding characteristic equation tend to zero as the small parameter $N^{−1}$ tends to zero. Known methods of local analysis based on the use of the method of invariant manifolds and the method of normal forms are not directly applicable in the problems under consideration. The main results consist in constructing so-called quasinormal forms — families of special nonlinear boundary value problems of parabolic type, which play the role of classical normal forms. Bifurcation phenomena are investigated and asymptotics of families of solutions bifurcating from the equilibrium state are constructed. The coupling parameter between chain elements and the parameter appearing in the boundary condition turn out to be fundamentally important. It is shown that the structure of solutions for nonperiodic chains is generally more complex than for periodic ones (ring chains).