Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

A simple example is considered of Hill's equation x" + (a^2 + bp(t))x = 0, where the forcing term p, instead of periodic, is quasi periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitrary b, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

Algunes reflexions sobre la matemàtica aplicada i l'ensenyament en general

Discurs inaugural del curs acadèmic 1996-199

Stability of parabolic points of area preserving analytic diffeomorphisms

Theorems characterizing stable parabolic points are proved. Essentially, stability is equivalent to the fact that the generating function of the differomorphism, taking out the part which generates the identity, has a strict extremum at the fixed point. With these results, the study of the stability of fixed points of analytic area preserving mappings (APM) is ended . Some examples are included, specially the case of elliptic points whose ei-genvalues are cubic or fourth roots of unit

Aplicacions quadràtiques que preserven l'àrea a R²

Considero la descripció, explicació i predicció de les propietats de les òrbites d'un sistema donat com un dels objectius principals dels sistemes dinàmics. En aquesta lliçó ens centrem en les aplicacions quadràtiques que preserven l'àrea (APM) a R2. Hi ha diverses raons per a aquesta elecció. En primer lloc, són un model paradigmàtic. A més, molts problemes referents a l'existència de corbes invariants difeomorfes a un cercle, el paper de les varietats invariants de punts fixos o periòdics de tipus hiperbòlic i com porten a l'existència de caos, els mecanismes geomètrics que porten a la destrucció de corbes invariants, i mesures quantitatives de les diferents propietats d'APM generals es poden entendre gràcies al nostre coneixement del cas quadràtic. En aquest article passem revista a alguns d'aquests temes. Al final es presenten diverses qüestions obertes i extensions a dimensió superior.I consider the description, explanation and prediction of the properties of the orbits of a given system as one of the main goals of Dynamical Systems. In this lecture we focus on the quadratic Area Preserving Maps (APM) in R2. There are several reasons for this choice. It is a paradigmatic model. Many problems concerning: the existence of invariant curves diffeomorphic to a circle; the role of invariant manifolds of hyperbolic fixed or periodic points and how they lead to the existence of chaos; the geometrical mechanisms leading to the destruction of invariant curves; and quantitative measures of different properties for general APM, can all be understood thanks to our knowledge of the quadratic case. A review of these topics is presented in the lecture. Several open questions and extensions are shown at the end of the lecture