Multi-clustered solutions for a forced pendulum equation
Institución: Universidad de Chile
Tipo de Evento: Investigación
We consider the singularly perturbed forced pendulum equation
\begin{equation*}
\varepsilon^2 u_{\varepsilon}^{\prime\prime}+\sin (u_{\varepsilon})=\varepsilon^2\alpha(t) u_{\varepsilon}+\varepsilon^2\beta(t) u_{\varepsilon}^{\prime}\qquad \text{in } (-L,L),
\end{equation*}
where \(\alpha,\beta\in C^2([-L,L],\mathbb{R})\) and \(u_{\varepsilon}\) represents the angle of the pendulum.
We shall present some recent results concerning the asymptotic behaviour of high energy solutions of this equation as the parameter \(\varepsilon\) approaches zero.
We shall also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof of these results relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.
This is a joint work with Salomé Martínez (Universidad de Chile).