Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method. II: Analytic case. (English) Zbl 1407.34094
Summary: We construct analytic quasi-periodic solutions of a state-dependent delay differential equation with quasi-periodically forcing. We show that if we consider a family of problems that depends on one dimensional parameters (with some non-degeneracy conditions), there is a positive measure set \(\Pi\) of parameters for which the system admits analytic quasi-periodic solutions.
The main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters. Our main result is proved by a Nash-Moser fast convergent method and is formulated in the a-posteriori format of numerical analysis. That is, given an approximate solution of a functional equation which satisfies some non-degeneracy conditions, we can find a true solution close to it.
This is in sharp contrast with the finite regularity theory developed in [J. Dyn. Differ. Equations 29, No. 4, 1503–1517 (2017; Zbl 1384.34080)]. We conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty. More precisely, for generic families of perturbations, the quasi-periodic solutions are only finitely differentiable in open sets in the complement of parameters set \(\Pi\).
The main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters. Our main result is proved by a Nash-Moser fast convergent method and is formulated in the a-posteriori format of numerical analysis. That is, given an approximate solution of a functional equation which satisfies some non-degeneracy conditions, we can find a true solution close to it.
This is in sharp contrast with the finite regularity theory developed in [J. Dyn. Differ. Equations 29, No. 4, 1503–1517 (2017; Zbl 1384.34080)]. We conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty. More precisely, for generic families of perturbations, the quasi-periodic solutions are only finitely differentiable in open sets in the complement of parameters set \(\Pi\).
MSC:
| 34K06 | Linear functional-differential equations |
| 34K27 | Perturbations of functional-differential equations |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 70K43 | Quasi-periodic motions and invariant tori for nonlinear problems in mechanics |
Keywords:
state-dependent delay; parameterization method; analytic solution; foliation-preserving torus map; KAM theory; a-posterioriCitations:
Zbl 1384.34080References:
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