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.

Polynomial normal forms of Constrained Differential Equations with three parameters

We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow-fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables.Comment: This is an updated and revised version. Minor modifications mad

Kolmogorov : la "k" de KAM

La teoria de Kolmogorov-Arnold-Moser (o kam) va ser desenvolupada per sistemes dinàmics conservatius que estan prop d’integrables. Típicament els sistemes integrables contenen molts tors invariants en el seu espai de fases. La teoria kam estableix resultats de persistència d’aquests tors, en els quals el moviment és quasiperiòdic. Fem un esbós d’aquesta teoria al voltant de la figura de Kolmogorov

A Global Kam-Theorem: Monodromy in Near-Integrable Perturbations of Spherical Pendulum

The KAM Theory for the persistence of Lagrangean invariant tori in nearly integrable Hamiltonian systems is lobalized to bundles of invariant tori. This leads to globally well-defined conjugations between near-integrable systems and their integrable approximations, defined on nowhere dense sets of positive measure associated to Diophantine frequency vectors. These conjugations are Whitney smooth diffeomorphisms between the corresponding torus bundles. Thus the geometry of the integrable torus bundle is inherited by the near-integrable perturbation. This is of intereet in cases where these bundles are nontrivial. The paper deals with the spherical pendulum as a leading example

Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts