Phase Reduction in the Noise Induced Escape Problem for Systems close to Reversibility

We consider n-dimensional deterministic flows obtained by perturbing a gradient flow. We assume that the gradient flow admits a stable curve of stationary points, and thus if the perturbation is not too large the perturbed flow also admits an attracting curve. We show that the noise induced escape problem from a stable fixed point of this curve can be reduced to a one-dimensional problem: we can approximate the associated quasipotential by the one associated to the restricted dynamics on the stable curve. The error of this approximation is given in terms of the size of the perturbation.Comment: 25 pages, 2 figure

Cavallero, Pablo Adrián. Leer a Homero. Ilíada, Odisea y la mitología griega. 1ª ed. Buenos Aires: Quadrata, 2014, 176 pp.

Fil: Poquet, Adriana. Universidad Nacional de Cuy

Transitions in active rotator systems: invariant hyperbolic manifold approach

Our main focus is on a general class of active rotators with mean field interactions, that is globally coupled large families of dynamical systems on the unit circle with non-trivial stochastic dynamics. Each isolated system is a diffusion process on a circle, with drift -delta V', where V' is a periodic function and delta is an intensity parameter. It is well known that the interacting dynamics is accurately described, in the limit of infinitely many interacting components, by a Fokker-Planck PDE and the model reduces for delta=0 to a particular case of the Kuramoto synchronization model, for which one can show the existence of a stable normally hyperbolic manifold of stationary solutions for the corresponding Fokker-Planck equation (we are interested in the case in which this manifold is non-trivial, that happens when the interaction is sufficiently strong, that is in the synchronized regime of the Kuramoto model). We use the robustness of normally hyperbolic structures to infer qualitative and quantitative results on the |delta|< delta0 cases, with delta0 a suitable threshold: as a matter of fact, we obtain an accurate description of the dynamics on the invariant manifold for delta=0 and we link it explicitly to the potential V . This approach allows to have a complete description of the phase diagram of the active rotators model, at least for |delta|< delta0, thus identifying for which values of the parameters (notably, noise intensity and/or coupling strength) the system exhibits periodic pulse waves or stabilizes at a quiescent resting state. Moreover, some of our results are very explicit and this brings a new insight into the combined effect of active rotator dynamics, noise and interaction. The links with the literature on specific systems, notably neuronal models, are discussed in detail.Comment: 29 pages, 4 figures. Version 2: some changes in introduction, added reference

César García Álvarez. La literatura clásica griega. Historia, textos, comentarios. Santiago de Chile, Editorial Universitaria, 2006, 180 pp.

Fil: Poquet, Adriana

Carlos García Gual. Encuentros heroicos. Seis escenas griegas. Madrid: FCE, 2009, 157 pp.

Fil: Poquet, Adriana. Universidad Nacional de Cuy

Graciela Maglia, María del Carmen Cabrero. Sintaxis de la decisión trágica. Sófocles. Áyax: Una lectura semiológica. Bogotá, Universidad Nacional de Colombia, 2005. 108 pp.

Fil: Poquet, Adriana