S-shaped bifurcations in a two-dimensional Hamiltonian system
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...
Elmentve itt :
Szerzők: | |
---|---|
Dokumentumtípus: | Folyóirat |
Megjelent: |
2021
|
Sorozat: | Electronic journal of qualitative theory of differential equations
|
Kulcsszavak: | Hamilton-rendszer, Bifurkáció |
doi: | 10.14232/ejqtde.2021.1.49 |
Online Access: | http://acta.bibl.u-szeged.hu/73701 |
LEADER | 01904nas a2200217 i 4500 | ||
---|---|---|---|
001 | acta73701 | ||
005 | 20211108155453.0 | ||
008 | 211108s2021 hu o 0|| eng d | ||
022 | |a 1417-3875 | ||
024 | 7 | |a 10.14232/ejqtde.2021.1.49 |2 doi | |
040 | |a SZTE Egyetemi Kiadványok Repozitórium |b hun | ||
041 | |a eng | ||
100 | 1 | |a Zegeling André | |
245 | 1 | 0 | |a S-shaped bifurcations in a two-dimensional Hamiltonian system |h [elektronikus dokumentum] / |c Zegeling André |
260 | |c 2021 | ||
300 | |a 38 | ||
490 | 0 | |a Electronic journal of qualitative theory of differential equations | |
520 | 3 | |a 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. | |
695 | |a Hamilton-rendszer, Bifurkáció | ||
700 | 0 | 1 | |a Zegeling Paul Andries |e aut |
856 | 4 | 0 | |u http://acta.bibl.u-szeged.hu/73701/1/ejqtde_2021_049.pdf |z Dokumentum-elérés |