S-shaped bifurcations in a two-dimensional Hamiltonian system
We study the solutions to the following Dirichlet boundary problem: \begin{equation*}\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0,\end{equation*} where $x \in \mathbb{R}$, $t \in \mathbb{R}$, $\lambda \in \mathbb{R}^+$, with boundary conditions: \begin{equation*} x(0)=x(1)=A \in \mathbb{R}. \end{equation...
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| Format: | Article |
| Language: | English |
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University of Szeged
2021-07-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8876 |
| _version_ | 1797830425472663552 |
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| author | Andre Zegeling Paul Zegeling |
| author_facet | Andre Zegeling Paul Zegeling |
| author_sort | Andre Zegeling |
| collection | DOAJ |
| description | We study the solutions to the following Dirichlet boundary problem:
\begin{equation*}\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0,\end{equation*}
where $x \in \mathbb{R}$, $t \in \mathbb{R}$, $\lambda \in \mathbb{R}^+$, with boundary conditions:
\begin{equation*}
x(0)=x(1)=A \in \mathbb{R}.
\end{equation*}
Especially we focus on varying the parameters $\lambda$ 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 $\lambda$ 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. |
| first_indexed | 2024-04-09T13:37:01Z |
| format | Article |
| id | doaj.art-2ba3b5df809e4bf9b4cdb3c255e79734 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:01Z |
| publishDate | 2021-07-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-2ba3b5df809e4bf9b4cdb3c255e797342023-05-09T07:53:11ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752021-07-0120214913810.14232/ejqtde.2021.1.498876S-shaped bifurcations in a two-dimensional Hamiltonian systemAndre Zegeling0Paul Zegeling1Guilin University of Aerospace Technology, Guilin, ChinaUtrecht University, Department of Mathematics, Budapestlaan 6, De Uithof, the NetherlandsWe study the solutions to the following Dirichlet boundary problem: \begin{equation*}\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0,\end{equation*} where $x \in \mathbb{R}$, $t \in \mathbb{R}$, $\lambda \in \mathbb{R}^+$, with boundary conditions: \begin{equation*} x(0)=x(1)=A \in \mathbb{R}. \end{equation*} Especially we focus on varying the parameters $\lambda$ 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 $\lambda$ 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.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8876ordinary differential equationsboundary value problemperiod function |
| spellingShingle | Andre Zegeling Paul Zegeling S-shaped bifurcations in a two-dimensional Hamiltonian system Electronic Journal of Qualitative Theory of Differential Equations ordinary differential equations boundary value problem period function |
| title | S-shaped bifurcations in a two-dimensional Hamiltonian system |
| title_full | S-shaped bifurcations in a two-dimensional Hamiltonian system |
| title_fullStr | S-shaped bifurcations in a two-dimensional Hamiltonian system |
| title_full_unstemmed | S-shaped bifurcations in a two-dimensional Hamiltonian system |
| title_short | S-shaped bifurcations in a two-dimensional Hamiltonian system |
| title_sort | s shaped bifurcations in a two dimensional hamiltonian system |
| topic | ordinary differential equations boundary value problem period function |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8876 |
| work_keys_str_mv | AT andrezegeling sshapedbifurcationsinatwodimensionalhamiltoniansystem AT paulzegeling sshapedbifurcationsinatwodimensionalhamiltoniansystem |