Dynamics of large nonperiodic chains of logistic equations with delay and unidirectional coupling

When considering the local dynamics of chains with a large number of elements, a transition is made from a discrete system of equations to a spatially distributed integrodifferential equation with boundary conditions. It is shown that critical cases in the problem of the stability of the equilibrium state have an infinite dimension in the sense that infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero. Infinite-dimensional analogs of normal forms—quasinormal forms—are constructed, which are special nonlinear parabolic boundary value problems of the Ginzburg–Landau type. Using these quasinormal forms, the dynamic properties of the solutions and the leading terms of their asymptotic expansions are determined. The existence of families of solutions that rapidly oscillate with respect to the spatial variable is established.