Large chains of delay logistic equations with advective-type constraints: an algorithm for constructing quasinormal forms

The dynamics of large non-periodic chains with advective connections between elements is considered. The main assumption is that the number $N$ of chain elements is sufficiently large, so a small parameter $\varepsilon=N^{-1}$ naturally arises. This assumption allows us to move from a system of $N$ delayed equations to the study of a spatially distributed integro-differential equation containing a small parameter and use asymptotic methods to investigate the dynamic properties of this equation. The connections between the chain elements are a difference approximation of the advection operator, which is why they are called advective. Another assumption is that the chains are not circular, i.e., the boundary conditions for the systems under consideration do not have periodic properties. Non-classical boundary conditions are considered, which lead to the emergence of new dynamic effects. Critical cases in the problem of equilibrium stability are identified, and it is shown that they have infinite dimension in the sense that an infinite number of roots of the characteristic equation approach the imaginary axis as a small parameter approaches zero. In this situation, the known research methods based on the use of invariant integral manifolds and normal forms are not directly applicable. We use methods of quasi-normal forms, whose non-local dynamics determine the local behavior of the solutions of the considered chains. The main results consist in constructing quasi-normal forms using special asymptotic methods. This allows us to obtain the main approximations of the solutions of the original chain with respect to the parameter $\varepsilon$.

dynamics, chains, bifurcations, stability, normal form, delay

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