We g i v e a proof based in geometric perturbation theory of a result proved by J.N. Mather using... more We g i v e a proof based in geometric perturbation theory of a result proved by J.N. Mather using variational methods. Namely, the existence of orbits with unbounded energy in perturbations of a generic geodesic ow i n T 2 by a generic periodic potential. amadeu@ma1.upc.es y llave@math.utexas.edu z tere@ma1.upc.es
The billiard motion inside an ellipsoid of R 3 is completely integrable. If the ellipsoid is not ... more The billiard motion inside an ellipsoid of R 3 is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geometrical, dynamical and topological point of view. It contains eight surfaces, called separatrices.
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal hete... more We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.
In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degr... more In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits.
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds interse... more For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect orbits with prescribed trajectories: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems.
Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler ... more Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626-1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo (1672) and displayed the results in triangular tables. In particular, his new arithmetic-algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli's view of his algorithmic computation of quadratures.
The splitting of separatrices for hyperbolic xed points of twist maps with d degrees of freedom i... more The splitting of separatrices for hyperbolic xed points of twist maps with d degrees of freedom is studied through a real-valued function, called the Melnikov p o t e n tial. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, Morse theory can beapplied to give lower bounds on the number of transverse homoclinic orbits.
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of ell... more We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptichyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel ′ nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
We g i v e a proof based in geometric perturbation theory of a result proved by J.N. Mather using... more We g i v e a proof based in geometric perturbation theory of a result proved by J.N. Mather using variational methods. Namely, the existence of orbits with unbounded energy in perturbations of a generic geodesic ow i n T 2 by a generic periodic potential. amadeu@ma1.upc.es y llave@math.utexas.edu z tere@ma1.upc.es
The billiard motion inside an ellipsoid of R 3 is completely integrable. If the ellipsoid is not ... more The billiard motion inside an ellipsoid of R 3 is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geometrical, dynamical and topological point of view. It contains eight surfaces, called separatrices.
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal hete... more We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.
In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degr... more In this paper we consider a representative a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits.
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds interse... more For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect orbits with prescribed trajectories: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems.
Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler ... more Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626-1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo (1672) and displayed the results in triangular tables. In particular, his new arithmetic-algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli's view of his algorithmic computation of quadratures.
The splitting of separatrices for hyperbolic xed points of twist maps with d degrees of freedom i... more The splitting of separatrices for hyperbolic xed points of twist maps with d degrees of freedom is studied through a real-valued function, called the Melnikov p o t e n tial. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, Morse theory can beapplied to give lower bounds on the number of transverse homoclinic orbits.
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of ell... more We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptichyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel ′ nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
Uploads
Papers by A. Delshams