| Autor: |
Paul Andries Zegeling, André Zegeling |
| Rok vydání: |
2021 |
| Předmět: |
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| Zdroj: |
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2021, Iss 49, Pp 1-38 (2021) |
| ISSN: |
1417-3875 |
| DOI: |
10.14232/ejqtde.2021.1.49 |
| Popis: |
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. |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
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