Dia: Dimecres 22 d'octubre de 2025

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

• Hora: 16h00m

A càrrec de: Dídac Gil Rams, UPC

Títol: Splitting of separatrices in generalized standard maps

Resum: We study transversal intersections between the invariant manifolds (stable and unstable) associated to an hyperbolic fixed point for a class of maps named generalized standard maps (defined in [1]). This generalization includes the Standard map, first studied by Lazutkin in [5], the perturbed McMillan map, studied in [6], or the Hénon map among others.

We obtain an asymptotic formula for the Lazutkin invariant, a value related to angle of intersection between the manifolds, which is exponentially small and its first term depends on a Stokes constant. To do so, one of the techniques that we use is the inner equation related to our generalized standard maps.

The Stokes constant is an involved quantity that comes from the study of the inner equation. The second part of our work consists in giving a general algorithm, useful for the studied maps, to compute an interval containing the value of the named constant by means of a computer assisted proof in CAPD (similarly as in [2]). Finally, we apply this algorithm to the Standard and Hénon maps in order to see that the intervals contain the values given by Gelfreich in [4] and [3], respectively.

Joint work with Inmaculada Baldomà Barraca (UPC), Maciej Capiński (AGH) and Pau Martín de la Torre (UPC).

Referències:

1. Baldomà, I., and Martín, P.: The inner equation for generalized standard maps.
2. Baldomà, I., Capiński, M. J., Giralt, M., and Guardia, M.: Breakdown of homoclinic orbits to l3: Nonvanishing of the stokes constant.
3. Gelfreich, V., and Sauzin, D.: Borel summation and splitting of separatrices for the Hénon map.
4. Gelfreich, V. G.: A Proof of the Exponentially Small Transversality of the Separatrices for the Standard Map.
5. Lazutkin, V.: Splitting of separatrices for the chirikov standard map.
6. Martín, P., Sauzin, D., and Seara, T. M.: Resurgence of inner solutions for perturbations of the Mcmillan map.

Last updated: Mon Oct 20 17:50:41 2025