Zegeling André; Zegeling Paul Andries: S-shaped bifurcations in a two-dimensional Hamiltonian system. (2021)
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We study the solutions to the following Dirichlet boundary problem: d 2x(t) dt2 + λ f(x(t)) = 0, where x ∈ R, t ∈ R, λ ∈ R+, with boundary conditions: x(0) = x(1) = A ∈ R. Especially we focus on varying the parameters λ and A in the case where the phase plane representation of the equation contains a saddle loop filled with a period annulus surrounding a center. We introduce the concept of mixed solutions which take on values above and below x = A, generalizing the concept of the well-studied positive solutions. This leads to a generalization of the so-called period function for a period annulus. We derive expansions of these functions and formulas for the derivatives of these generalized period functions. The main result is that under generic conditions on f(x) so-called S-shaped bifurcations of mixed solutions occur. As a consequence there exists an open interval for sufficiently small A for which λ can be found such that three solutions of the same mixed type exist. We show how these concepts relate to the simplest possible case f(x) = x(x + 1) where despite its simple form difficult open problems remain.
| Mű típusa: | Folyóirat |
|---|---|
| Folyóirat/könyv/kiadvány címe: | Electronic journal of qualitative theory of differential equations |
| Dátum: | 2021 |
| Szám: | 49 |
| ISSN: | 1417-3875 |
| Oldalszám: | 38 |
| Nyelv: | angol |
| DOI: | 10.14232/ejqtde.2021.1.49 |
| Kulcsszavak: | Hamilton-rendszer, Bifurkáció |
| Megjegyzések: | Bibliogr.: p. 36-38. ; összefoglalás angol nyelven |
| Feltöltés dátuma: | 2021. nov. 08. 15:54 |
| Utolsó módosítás: | 2021. nov. 08. 15:54 |
| URI: | http://acta.bibl.u-szeged.hu/id/eprint/73701 |
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