Rafael Ortega

18071 Granada, Spain
Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada


Gebhard B., Ortega R.
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 whole-plane 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.
Keywords: vortex dynamics, periodic solutions, stability, Floquet multipliers, bifurcation, Poincaré section
Citation: Gebhard B., Ortega R.,  Stability of Periodic Solutions of the $N$-vortex Problem in General Domains, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 649-670
Ortega R.
Stable Periodic Solutions in the Forced Pendulum Equation
2013, vol. 18, no. 6, pp.  585-599
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.
Keywords: Lyapunov stability, forced pendulum, prevalence, periodic solution, regular value, discriminant
Citation: Ortega R.,  Stable Periodic Solutions in the Forced Pendulum Equation, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 585-599

Back to the list