Papers by Amadeu Delshams
Nonlinearity, 2015
We study bifurcations of area-preserving maps, both orientable (symplectic) and nonorientable, wi... more We study bifurcations of area-preserving maps, both orientable (symplectic) and nonorientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.
Regular and Chaotic Dynamics, 2014
Introduction At the end of the last century, H. Poincare [7] discovered the phenomenon of separat... more Introduction At the end of the last century, H. Poincare [7] discovered the phenomenon of separatrices splitting, which now seems to be the main cause of the stochastic behavior in Hamiltonian systems. He formulated a general problem of Dynamics as a perturbation of an integrable Hamiltonian system H(I; '; ") = H 0 (I) + "H 1 (I; '); where " is a small parameter, I = (I 1 ; I 2 ; : : : ; I n ), ' = (' 1 ; ' 2 ; : : : ; ' n ). The values of the actions I, such that the unperturbed frequencies ! k (I) = @H 0 =@I k are rationally dependent, are called resonances. As a model for the
Mathematical Physics Electronic Journal
The splitting of separatrices for Hamiltonians with $1{1\over 2}$ degrees of freedom $$ h(x,t /\v... more The splitting of separatrices for Hamiltonians with $1{1\over 2}$ degrees of freedom $$ h(x,t /\varepsilon ) = h^{0}(x) + \mu \varepsilon ^{p} h^{1}(x,t /\varepsilon ) $$ is measured. We assume that $ h^{0}(x)= h^{0}(x_{1},x_{2})= x_{2}^{2}/2+V(x_{1})$ has a separatrix $x^{0}(t)$, $ h^{1}(x,\theta )$ is $2\pi $-periodic in $\theta $, $\mu $ and $\varepsilon >0 $ are independent small parameters, and $p\ge 0$. Under suitable conditions of meromorphicity for $x_{2}^{0}( u )$ and the perturbation $ h^{1}(x^{0}( u ),\theta )$, the order $ \ell $ of the perturbation on the separatrix is introduced, and it is proved that, for $ p \ge \ell $, the splitting is exponentially small in $\varepsilon $, and is given in first order by the Melnikov function.
The IMA Volumes in Mathematics and its Applications, 2001
We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori wit... more We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers, putting emphasis on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori. A geometric method is presented whichtak es into accountt h e Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: the Lyapunov exponents of the whiskered tori remain xed), the splitting potential is wellapproximated by a Melnikov potential. This method is designed as a rst step in the study of the singular case (also known as a priori-stable: the Lyapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).
Regular and Chaotic Dynamics, 2014
We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori ... more We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number Ω = √ 2 − 1. We show that the Poincaré -Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter ε satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of ε, generalizing the results previously known for the golden number.
Electronic Research Announcements in Mathematical Sciences, 2014
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in ... more We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems. We consider 2-dimensional tori with a frequency vector ω = (1, Ω) where Ω is a quadratic irrational number, or 3-dimensional tori with a frequency vector ω = (1, Ω, Ω 2 ) where Ω is a cubic irrational number. Applying the Poincaré-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Ω is the so-called cubic golden number (the real root of x 3 + x − 1 = 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.
International Journal of Bifurcation and Chaos, 2014
We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyint... more We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector ω/ √ ε, with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré-Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.
Communications in Mathematical Physics, 2000
Wegive a proof based in geometric perturbation theory of a result proved by J.N. Mather using var... more Wegive 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 owinT 2 by a generic periodic potential. [email protected] y [email protected] z [email protected]
(In 2 Volumes), 2000
The splitting of separatrices of area preserving maps close to the identity is one of the most pa... more The splitting of separatrices of area preserving maps close to the identity is one of the most paradigmatic examples of an exponentially small or singular phenomenon. The intrinsic small parameter is the characteristic exponent h > 0 of the saddle fixed point. A standard technique to measure the splitting of separatrices is the so-called Poincaré-Melnikov method, which has several specific features in the case of analytic planar maps. The aim of this talk is to compare the predictions for the splitting of separatrices provided by the Poincaré-Melnikov method, with the analytic and numerical results in a simple example where computations in multiple-precision arithmetic are performed.
Hamiltonian Systems with Three or More Degrees of Freedom, 1999
Nonlinear Science and Complexity, 2011
Page 128. An Accounting Device for Biasymptotic Solutions: The Scattering Map in the Restricted T... more Page 128. An Accounting Device for Biasymptotic Solutions: The Scattering Map in the Restricted Three Body Problem Amadeu Delshams, Josep J. Masdemont1 and Pablo Roldán Abstract We compute the scattering map (see ...
We study dynamics and bifurcations of two-dimensional reversible maps having a symmetric saddle f... more We study dynamics and bifurcations of two-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figureeight). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.
The (planar) ERTBP describes the motion of a massless particle (a comet) under the gravitational ... more The (planar) ERTBP describes the motion of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show that there exist trajectories of motion such that their angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive trajectories, that is, with a large variation of angular momentum.
We consider models given by Hamiltonians of the form H(I, ϕ, p, q, t; ε) = h(I)+ n j=1 60 Referen... more We consider models given by Hamiltonians of the form H(I, ϕ, p, q, t; ε) = h(I)+ n j=1 60 References 60
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Papers by Amadeu Delshams