Amadeu Delshams

Universitat Politècnica de Catalunya | UPC · Department of Applied Mathematics I (MA1)

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector \(\omega /\sqrt{\varepsilon }\), with \(\omega =(1,\Omega ,\widetilde{\Omega })\) where \(\Omega \...

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega,\widetilde\Omega)$ where $\Omega$ is a cubic irr...

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 angul...

We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that Gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasi-pe...

We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that Gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasi-pe...

We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the...

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact man...

We present a new result about the shadowing of nontransversal chain of heteroclinic connections based on the idea of dropping dimensions. We illustrate this new mechanism with several examples. As an application we discuss this mechanism in a simplification of a toy model system derived by Colliander et al. in the context of cubic defocusing nonlin...

We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional manifolds. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits.

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p 2/2+ cos q − 1 + I 2/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a 00 + a 10 cosφ + a 01 cos s) (a 10 a 01 ≠ 0) there is glo...

We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L1, located between Sun and Earth. Near L1 there exists a normally hyperbo...

We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at...

The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega)$ where the frequency ratio $\Omega$ is a quadratic irrationa...

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a two-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbat...

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely...

We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.

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 $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be appli...

We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, 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-s...

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector $\omega/\sqrt\epsilon$, with $\omega=(1,\Omega)$ where $\Omega$ is an irrational number of constant type,...

We consider models given by Hamiltonians of the form $$H(I,\phi,p,q,t;\epsilon) = h(I) + \sum_{j = 1}^n \pm(\frac{1}{2} p_j^2 + V_j(q_j)) + \epsilon Q(I,\phi,p,q,t;\epsilon)$$ where $I,\phi$ are d-dimensional actions and angles, $p,q$ are n-dimensional real conjugated variables, and $t$ is an angle. These are higher dimensional analogues, both in t...

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 $\omega=(1,\Omega)$ where $\Omega$ is a quadratic irrational number, or 3-dimensional tori with a frequency vector $\omega=(1,\Omega,\Omega^2)$ where $\Omega$...

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive a...

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive a...

We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that t...

For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: 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...

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...

We introduce a geometric mechanism for di?usion in a priori unstable nearly integrable dynamical systems. It is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. We argue that these objects created by resonances can be incorporated in transition chains taking the pl...

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that qua...

We compute the scattering map (see explanation below) in the Spatial Restricted Three Body Problem using a combination of analytical and numerical techniques.

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 $\v...

La famosa memòria sobre el problema de tres cossos que Poincaré presentà en juny del 1988 al concurs commemoratiu dels 60 anys del Rei Òscar de Suècia va rebre el premi el 20 de gener del 1989. Ara bé, la primera versió presentada a l'Acta Mathematica contenia un error essencial, ja que, expressant'ho en el llenguatge actual dels sistemes dinàmics,...

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom and we apply the geometric mechanism for diffusion introduced in [A. Delshams, R. de la Llave, T.M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on...

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center...

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic– hyperbolic 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 m...

In this paper we consider the case of a general perturbation, for r large enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom, and we provide explicit conditions on it, which turn out to be generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion. This is a generalization of the...

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, 70years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geome...

Let �1 and �2 be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of �1 intersects the unstable manifold of �2 transversally along a manifold . The scattering map from � 2 to �1 is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heterocli...

We present (informally) some geometric structures that imply instability in Hamiltonian systems. We also present some finite calculations which can establish the presence of these structures in a given near integrable systems or in systems for which good numerical information is available. We also discuss some quantitative features of the diffusion...

Given a normally hyperbolic invariant manifold Λ for a map f, whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future.We show that when f and Λ are symplectic (respectively exact symplectic...

A general theory for perturbations of an integrable planar map with a separatrix to a hyperbolic fixed point has been developed in a previous lecture [5]. The splitting of the perturbed invariant curves was measured, in first order with respect to the parameter of perturbation, by means of a periodic Melnikov function M defined on the unperturbed s...

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the inform...

Homoclinic orbits for invariant tori of nearly integrable Hamitonian systems, were studied. It was assumed that Hamiltonian vector field has a simple singular point O of type center-center-saddle. The spectrum of linearization operator at the point O was assumed to have two pairs of purely imaginary eigenvalues and a pair of nonzero real values. It...

We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:(a)The metric and the external perturbation are smooth enough.(b)The geodesic flow has a hyperbolic periodic orbit such that...

Homoclinic and heteroclinic connections between planar Lyapunov orbits of the Sun-Earth and Earth-Moon models can be found by using their hyperbolic invariant manifolds and Poincare section representations. These connections can be classified in bifurcation families according to the range of values of the associated Jacobi constant. In the formalis...

We study the existence of transverse homoclinic orbits in a singular or weakly hyperbolic Hamiltonian, with $3$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The example considered consists of an integrable Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast fr...

We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center–center–saddle equilibrium having a homoclinic orbit or loop. With the help of a Poincaré map (chosen based on the unperturbed homoclinic loop), we study the homoclinic intersections between the stable and unstable manifolds associated to...

In this paper we introduce the pseudo-normal form, which generalizes the notion of normal form around an equilibrium. Its convergence is proved for a general analytic system in a neighborhood of a saddle-center or a saddle-focus equilibrium point. If the system is Hamiltonian or reversible, this pseudo-normal form coincides with the Birkhoff normal...

We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing an $n$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whisk...

We consider an example of singular or weakly hyperbolic Hamiltonian, with 3 degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast frequencies omega/(epsilon^(1/2)) and coincident wh...

Con motivo del sesquicentenario del nacimiento de Henri Poincaré, resulta impresionante comprobar la influencia actual de su obra, así como el gran adelanto de sus métodos e ideas respecto a las de los científicos coetáneos. En esta conferencia se repasan algunas de sus contribuciones principales a las ecuaciones diferenciales y ala mecánica celest...

A mathematical model of cotransport phenomena of metal ions across flat sheet-supported liquid membranes (FSSLM) is developed in the transient. The initial conditions of the FSSLM system, the physical characteristics of the membrane (porosity, tortuosity and thickness), the kinetics of the chemical reaction in the feed–membrane interface and the di...

We consider the billiard motion inside a C²-small perturbation of an n-dimensional ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are doubled, so that there is an n-dimensional invariant set W of homoclinic orbits for the unperturbed billiar...

We present a geometric mechanism for diffusion in Hamiltonian systems. We also present tools that allow to verify it in a concrete model. In particular, we verify it in a system which presents the large gap problem.

A mathematical model is developed for the carrier facilitated transport of metal ions through a flat sheet support liquid membrane (FSSLM) in transition state from Fick’s second law. From this model, and from Fick’s first law, the flow density is derived as a non-linear concentration gradient. Both expressions, concentration and flow density, depen...

Peer reviewed

The billiard motion inside an ellipsoid of 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. The splitting of the separatri...

We consider perturbations of integrable, area preserving nontwist maps of the annulus (those are maps in which the twist condition changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves. We show in suitable two-parameter families the persistence of critical circles (invariant circles whose rotat...

At the end of the last century, H. Poincaré [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 where e is a small parameter, I = (I 1,I 2,…., I n), = The valu...

this paper this result is good enough only for the regular case. However, it is expected that this approximation also holds in the singular case = "

We give 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 flow in ު by a generic periodic potential.

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-M...

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and nonintegrability is stressed and in particular it is proved that quad...

ion, the stable and unstable invariant curves of the perturbed map intersect transversally along exactly two primary homoclinic orbits in the first quadrant; in particular, the unperturbed separatrix splits. The term primary means that the homoclinic orbits persist for all " small enough. The pieces of the perturbed invariant curves between two con...

We consider families of analytic area-preserving maps depending on two pa- rameters: the perturbation strength E and the characteristic exponent h of the origin. For E=0, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as h->0+. For fixed E!=0 and small h, we show that these con...

Peer reviewed

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...

this paper this result is good enough only for the regular case. However, it is expected that this approximation also holds in the singular case = "

We consider a perturbation of an integrable Hamiltonian system, possessing hyperbolic invariant tori with coincident whiskers. Following an idea due to Eliasson, we introduce a splitting potential whose gradient gives the splitting distance between the perturbed stable and unstable whiskers. The homoclinic orbits to the perturbed whiskered tori are...

this paper. The aim of the following sections is to show that, using suitable variables, the "whole" splitting distance (and not only its first order approximation) is the gradient of some function, in order to establish the existence of homoclinic orbits even in the singular case. In section 4, we introduce flow-box variables in some real neighbor...

RESUMEN We develop a method to study resonances for three-dimensional frequency vectors ! in the way that was done in (1, 2) for the two-dimensional case. We consider the case of cubic frequency vectors of the form ! = (1;›;›2), where › is an irrational root of a polynomial p(x) of degree 3, with integer coe-cients. In the complex case (when p(x) h...

. We consider perturbations of integrable, area preserving non-twist maps of the annulus (those are maps that violate the twist condition in a very strong sense: @q 0 =@p changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves. We show in suitable 2-parameter families the persistence of critical...

We give 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 flows in T 2 by a generic periodic potential. amadeu@ma1.upc.es y llave@math.utexas.edu z tere@ma1.upc.es 1 1 INTRODUCTION 2 1 Introduction T...

It is difficult, in the theory of dynamical systems, to draw a boundary line between conservation laws and symmetries because often their effects on the dynamics can be almost the same. An important example can be the case of motions in Hamiltonian and reversible systems. For instance, it is well-known that the Kolmogorov-Arnol’d-Moser theory appli...

We give a precise statement of KAM theorem for a Hamiltonian system in a neighborhood of an elliptic equilibrium point. If the frequencies of the elliptic point satisfy a Diophantine condition, with exponent $\tau$, and a nondegeneracy condition is fulfilled, we show that in a neighborhood of radius $r$ the measure of the complement of the KAM tori...

The splitting of separatrices for the standard-like maps $$F(x,y) = \left( {y, - x + \frac{{2\mu y}} {{1 + y^2 }} + \varepsilon V'(y)} \right), \mu = cosh h, h > 0, \varepsilon \in \mathbb{R},$$ is measured. For even entire perturbative potentials V(y) = Σn≥2V n y 2n such that \(\hat V(2\pi ) \ne 0\), where \(\hat V(\xi ) = \sum\nolimits_{n \geqsla...

IntroductionA general theory for perturbations of an integrable planar map with a separatrix to ahyperbolic fixed point has been developed in a previous lecture [5]. The splitting of theperturbed invariant curves was measured, in first order with respect to the parameter ofperturbation, by means of a periodic Melnikov function M defined on the unpe...

The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given....

The splitting of separatrices for Hamiltonians with 1 1 2 degrees of freedom h(x; t=") = h 0 (x) + " p h 1 (x; t=") 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; `) is 2-periodic in `, and " ? 0 are independent small parameters, and p 0. Under suitable conditions of meromorphicity for...

Introduction A century ago, the phenomenon of the splitting of separatrices was discovered by Henri Poincar'e in his celebrated memoir on the three-body problem [39]. While trying to integrate the problem of the three bodies, expanding the solutions with respect to a small parameter, Poincar'e noticed that the main obstruction was due to the possib...