Rafael Ortega
18071 Granada, Spain
Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada
Publications:
Gebhard B., Ortega R.
Stability of Periodic Solutions of the $N$vortex Problem in General Domains
2019, vol. 24, no. 6, pp. 649670
Abstract
We investigate stability properties of a type of periodic solutions of the $N$vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the wholeplane vortex system and a critical point $a_0\in\Omega$ of the Robin function associated to the Dirichlet Laplacian of $\Omega$. Under a linear stability condition on the initial rotating configuration, which can be verified for
examples
consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. If $a_0$ is a nondegenerate maximum or minimum, they are stable in a certain linear sense. Since nondegenerate minima exist generically, our results apply to most domains $\Omega$. The influence of the general domain $\Omega$ can be seen as a perturbation breaking the symmetries of the $N$vortex system on $\mathbb{R}^2$. Symplectic reduction is not applicable and our analysis on linearized stability relies on the notion of approximate eigenvectors. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.

Ortega R.
Stable Periodic Solutions in the Forced Pendulum Equation
2013, vol. 18, no. 6, pp. 585599
Abstract
Consider the pendulum equation with an external periodic force and an appropriate condition on the length parameter. It is proved that there exists at least one stable periodic solution for almost every external force with zero average. The stability is understood in the Lyapunov sense.
