The Poincaré-Melnikov-Arnold method for planar maps gives rise to a Melnikov function defined by an infinite and (a priori) analytically uncomputable sum. Under an assumption of meromorphicity, residues theory can be applied to provide an equivalent finite sum. Moreover, the Melnikov function turns out to be an elliptic function and a general criterion about non-integrability is provided. Several examples are presented with explicit estimates of the splitting angle. In particular, the non-integrability of non-trivial symmetric entire perturbations of elliptic billiards is proved, as well as the non-integrability of standard-like maps.

This paper is concerned with establishing broadly-based system-theoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity; and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory The role of structural constraints such as causality, locality, finite memory, etc., are then discussed as rules of the game. These concepts and results are applied to the specific problem or stochastic automaton, or grammar, inference. Computational results are given to demonstrate that the theory is complete and fully operational. Finally the formulation of identification proposed in this paper is analysed in terms of Klir’s epistemological hierarchy and both are discussed in terms of the rich philosophical literature on the acquisition of knowledge. 1

Abstract. The classical Arnold-Liouville theorem describes the geometry of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a connected singular nondegenerate level set, after a normal finite covering, admits a non-complete system of action-angle functions (the number of action functions is equal to the rank of the moment map), and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities. These results are essential for the global topological study of integrable Hamiltonian systems. 1.

We give a precise statement for KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain order K 4, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the complement of the KAM tori in a neighbourhood of given radius. Moreover, if the frequencies satisfy a Diophantine condition, with exponent , we show that in a neighbourhood of radius r the measure of the complement is exponentially small in (1=r) 1=(+1) . We also give a related result for quasi-Diophantine frequencies, which is more useful for practical purposes. The results are obtained by putting the system in Birkhoff normal form up to an appropiate order, and the key point relies on giving accurate bounds for its terms.

Abstract. We prove that Birkhoff normal form of hamiltonian flows at a nonresonant singular point with given quadratic part are always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral.

. We present new proofs of two results on the billiard ball problem by Rychlik [R] and Bialy [B]. x0. INTRODUCTION. We will give new proofs of two results on the billiard ball problem by Rychlik [R] and Bialy [B]. The original proofs were based on variational considerations. In our approach the variational context is absent, the dynamical system takes the center stage. We hope the simplifications provided by our method will make possible some progress on the conjectures for which these results lend partial support. x1.THE DYNAMICAL SYSTEM. Let us consider a convex domain Q in the plane. The billiard ball system is the flow \Phi t on Q \Theta S 1 defined by the free motion of a point particle in Q, with elastic reflections at the boundary @Q (the angle of reflection equal to the angle of incidence). The circle S 1 represents unit velocities. Strictly speaking, we need to identify the velocities at the boundary according to the collision law. The flow \Phi t preserves the Liou...

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when M is a homotopy 3-sphere, the width is loosely speaking the

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We perform a bifurcation analysis of normal-internal resonances in parametrised families of quasi-periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the `backbone' system; forced, the system is a skew-product flow with a quasi-periodic driving with n basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The averaged system turns out to have the same structure as in the well-known case of periodic forcing (n = 1); for a real analytic system, the non--integrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi-periodic n-dimensional tori in the averaged system, filling normal-internal resonance `gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of `gaps within gaps' makes the quasi--periodic case more complicated than the periodic case.

Let T ⊂ Rm+1 be a strictly convex domain bounded by a smooth hypersurface X = ∂T. In this paper we find lower bounds on the number of billiard trajectories in T which have a prescribed initial point A ∈ X, a prescribed final point B ∈ X, and make a prescribed number n of reflections at the boundary X. We apply a topological approach based on the calculation of cohomology rings of certain configuration spaces of Sm.