Chains with diffusion-type couplings contaning a large delay. (English. Russian original) Zbl 07878653
Math. Notes 115, No. 3, 323-335 (2024); translation from Mat. Zametki 115, No. 3, 355-370 (2024).
Summary: We investigate the local dynamics of a system of oscillators with a large number of elements and with diffusion-type couplings containing a large delay. We isolate critical cases in the stability problem for the zero equilibrium state and show that all of them are infinite-dimensional. Using special infinite normalization methods, we construct quasinormal forms, that is, nonlinear boundary value problems of parabolic type whose nonlocal dynamics determines the behavior of solutions of the original system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of dynamic properties of the original problem.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35K58 | Semilinear parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R09 | Integro-partial differential equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
Keywords:
periodic boundary conditions; delay; stability; normal form; dynamics; asymptotics of solutionsReferences:
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