Delshams i Valdes, Amadeu
Total activity: 367
h index
19
Professional category
University professor
Doctoral courses
Matemàtiques
Premio extraordinario
University degree
Física
Matemàtiques
Research group
EGSA - Differential Equations, Geometry, Control and Dynamical Systems, and Applications
Department
Department of Applied Mathematics I
School
Barcelona School of Industrial Engineering (ETSEIB)
E-mail
amadeu.delshamsupc.edu
Contact details
UPC directory
Orcid
0000-0003-4134-8882
Scopus Author ID
6701671914

## Scientific and technological production Ordered by:  Date asc. Date desc. Title asc. Title desc. Researcher asc. Researcher desc.

1 to 50 of 367 results
• Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Electronic research announcements in mathematical sciences
Date of publication: 2014-01-01
Journal article

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show 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 Q 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.

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show 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 Q 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.

• Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Date: 2014-02
Report

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 $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincar´e¿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.

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 $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincar´e–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.

• Global instability in the elliptic restricted three body problem

de la Rosa Ibarra, Abraham
Defense's date: 2014-06-16
Universitat Politècnica de Catalunya
Theses

El objetivo de esta tesis es mostrar inestabilidad global o difusión de Arnold en el problema restringido de tres cuerpos elíptico (PTCRE) mostrando la existencia de pseudo-trayectorias difusivas en el espacio fase para ciertos rangos de la excentricidad (e), el momento angular del cometa (G) y el parámetro de masa (µ). Mas precisamente, los resultados presentados, son válidos para G suficientemente grande, eG acotado y µ suficientemente pequeño.La tesis está dividida en dos capítulos y dos apéndices. El capítulo 1, contiene todos los resultados principales. Después de introducir el PTCRE, usamos coordenadas de McGehee para definir la variedad de infinito, que será de dimensión tres en el espacio fase extendido y que topológicamente se comporta como una variedad invariante normalmente hiperbólica (NHIM), aunque es de tipo parabólico. Esto significa que la tasa de acercamiento y alejamiento de ella a lo largo de sus variedades invariantes es polinomial, en lugar de exponencial como sucede en una NHIM estándar. Por otra parte, la dinámica interior es trivial ya que está formada por una familia de orbitas con 2 parámetros y de período 2p en el espacio extendido 5D que corresponden a soluciones constantes en el espacio reducido 4D. Como consecuencia, las variedades estables e inestables de la variedad de infinito son la unión de las variedades estables e inestables de sus orbitas periódicas y siempre que estas variedades se intersequen sobre orbitas heteroclínicas transversales, el scattering map puede ser definido como hicieron De la Llave, Seara y Delshams . Desafortunadamente, ya que la dinámica interior de la variedad de infinito es muy simple, el mecanismo de difusión clásico, que consiste en combinar la dinámica interior con la exterior, no funciona aquí. En su lugar, como una novedad, seremos capaces de encontrar dos scattering maps diferentes que serán combinados de manera adecuada para producir orbitas cuyo momento angular crezca.La fórmula asintótica del scattering map recae enteramente en el cálculo del llamado potencial de Melnikov, como es definido en los trabajos de Delshams, Gutiérrez y Seara. La primer derivada del potencial de Melnikov da la aproximación a primer orden de la distancia entre las variedades estable e inestable de la variedad de infinito cuando el parámetro de masa es exponencialmente pequeño. Con este planteamiento, una serie de lemas y proposiciones conducirán a la fórmula de los términos dominantes del potencial de Melnikov. La idea clave es calcular sus coeficientes de Fourier, que serán exponencialmente pequeños cuando el momento angular es grande y una fórmula explícita no será posible, así que un cálculo efectivo será necesario. Para hacerlo, el producto eG jugará un papel clave que conducirá a los teoremas 1.5 y 1.6, el primero da una fórmula asintótica del potencial de Melnikov cuando eG es pequeño y el segundo cuando eG es finito. Ambos requieren que µ sea exponencialmente pequeño con respecto a G, y G suficientemente grande. Estos teoremas naturalmente producirán las fórmulas asintóticas de los scattering maps para ambos casos y son la base de los teoremas 1.15 y 1.16, que formulan la existencia de pseudo-trayectorias en el PTCRE.En el capítulo 2, damos los detalles y las pruebas de los resultados concernientes a las formulas asintóticas, dadas en el capítulo 1, para el potencial de Melnikov y los scattering maps, incluyendo las cotas efectivas de cada error involucrado. Los apéndices tienen los resultados mas técnicos que son necesarios para completar de forma rigurosa cada prueba, pero que por su naturaleza, pueden ser relegados al final para hacer seguir las pruebas con mas facilidad.

• Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Delshams i Valdes, Amadeu; Gonchenko, Sergey; GONCHENKO, VLADIMIR; Lazaro Ochoa, Jose Tomas; Stenkin, Oleg
Nonlinearity
Date of publication: 2013-01-01
Journal article

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle fixed points. We consider one-parameter families of reversible maps unfolding the initial heteroclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations and birth of asymptotically stable, unstable and elliptic periodic orbits

• Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
Physica. D, Nonlinear phenomena
Date of publication: 2013-01-15
Journal article

• Exponentially small splitting of separatrices for 2D and 3D invariant tori

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Gonchenko, Marina
International Conference Dynamics, Bifurcations and Strange Attractors
Presentation's date: 2013-07-01
Presentation of work at congresses

• Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Date: 2013
Report

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 $\Omega$ is a cubic irrational number. Applying the Poincar´e¿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 $\Omega$ 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.

We study the splitting of invariant manifolds of whiskered t ori 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 v ector ω = (1 , Ω , Ω 2 ) where Ω is a cubic irrational number. Applying the Poincar ́e–Melnikov method, we find exponentia lly small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associa ted to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequen cies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfille d in 24 cases, which allows us to provide asymptotic estimate s in a simple way. In the cubic case, we focus our attention to th e 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 cubi c cases.

• Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion

Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
Date: 2013-06
Report

We consider models given by Hamiltonians of the form $$H(I,\varphi,p,q,t;\varepsilon) = h(I)+\dsum^n_{j=1}\pm(\frac12p^2_j+V_j(q_j))+\varepsilon Q(I,\varphi,p,q,t;\varepsilon)$$, where $I \in {\cal I}\subset \mathbb{R}^d$, $\varphi\in\mathbb{T}^d$, $p, q\in\mathbb{R}^n$, $t\in\mathbb{T}^1$. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All these models present the large gap problem. We show that, for $0< \varepsilon << 1$, under regularity and explicit nondegeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. The method of proof follows closely the strategy of [DLS03, DLS06a]. The main new phenomenon that appears when the dimension d of the center directions is larger than one, is the existence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [DLS03, DLS06a]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS08], not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS03, DLS06a].

Abstract. We consider models given by Hamiltonians of the form H ( I;';p;q;t ; " ) = h ( I )+ n X j =1 1 2 p 2 j + V j ( q j ) + "Q ( I;';p;q;t ; " ) where I 2I R d ;' 2 T d , p;q 2 R n , t 2 T 1 . These are higher di- mensional analogues, both in the center and hyperbolic directions, of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All these models present the large gap problem . We show that, for 0 < " 1, under regularity and explicit non- degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O (1). This domain includes resonance lines and, hence, large gaps among d -dimensional KAM tori. The method of proof follows closely the strategy of [DLS03, DLS06a]. The main new phenomenon that appears when the di- mension d of the center directions is larger than one, is the exis- tence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I , they can be contoured. This corresponds to the mechanism called di usion across resonances in the Physics literature. The present paper, however, di ers substantially from [DLS03, DLS06a]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS08], not avail- able when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the res- onances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS03, DLS06a]

• Homoclinic phenomena in conservative systems

Gonchenko, Marina
Defense's date: 2013-04-29
Universitat Politècnica de Catalunya
Theses

L¿objectiu d¿aquesta tesi és l¿estudi d¿òrbites homoclíniques en sistemes conservatius (aplicacions que conserven àrea i sistemes hamiltonians). Considerem òrbites homoclíniques (bi-asimptòtiques) o bé a òrbites periòdiques de tipus sella, o bé a tors amb bigotis. Aquestes òrbites, anomenades homoclíniques per Poincaré, són de gran interès ja que la seva presència implica dinàmica complicada. La tesi està dividida en dues parts segons els temes tractats. A la primera part, s'estudien les aplicacions que conserven àrea (APMs) amb una òrbita homoclínica no transversal (tangència homoclínica) a un punt fix de tipus sella. L¿objectiu és conèixer el comportament de les òrbites a prop de la trajectòria homoclínica donada. Per això, construïm aplicacions ¿de primer retorn¿, per a les quals fem servir formes normals finitament diferenciables de les aplicacions de tipus sella i introduïm coordenades creuades. Els punts fixos de les aplicacions ¿de primer retorn¿ corresponen a òrbites periòdiques de primera volta de les aplicacions considerades. Utilitzant mètodes de rescaling, reduïm les aplicacions ¿de primer retorn¿ a aplicacions de tipus Hénon, les bifurcacions de les quals són ben conegudes. Per tant, traslladant els resultats obtinguts per als punts fixos de les aplicacions ¿de primer retorn¿ a les òrbites periòdiques, provem l'existència de cascades de punts periòdics el¿líptics. També estudiem el fenomen de la coexistència d¿un nombre infinit d'òrbites periòdiques de diferents períodes (anomenada ressonància global). Considerem els problemes relacionats de diferents tipus de APMs (aplicacions simplèctiques i APMs no orientables) amb tangències quadràtiques o cúbiques. També establim l'estructura de ressonància 1:4 per algunes aplicacions de tipus Hénon conservatives.A la segona part de la tesi, estudiem l¿escissió (splitting) de separatrius exponencialment petita obtinguda a partir d'una pertorbació d'un sistema hamiltonià amb una connexió homoclínica (separatriu). Considerem una pertorbació d'un sistema hamiltonià integrable que té tors amb bigotis (hiperbòlics) amb els bigotis (varietats invariants) estables i inestables coincidents. Generalment, al sistema pertorbat ja no coincideixen els bigotis i el nostre objectiu és detectar les òrbites homoclíniques transversals associades al tor persistent. El sistema pertorbat resulta ser no integrable a causa de la presència d'aquestes trajectòries homoclíniques i, per tant, hi ha una dinàmica caòtica a prop seu. Donem una parametrització dels bigotis apropiada per tal de determinar la distància entre ells. Aquesta distància es coneix com funció de splitting, i els zeros simples d¿aquesta funció donen lloc a òrbites homoclíniques transversals. Fem servir el mètode clàssic de Poincaré-Melnikov per mesurar l¿escissió, encara que en el cas exponencialment petit hem de garantir que l'aproximació de primer ordre supera el terme d'error. Considerem sistemes hamiltonians que tenen tors amb bigotis bidimensionals amb freqüències quadràtiques i tors amb bigotis tridimensionals amb freqüències cúbiques. En el cas de dimensió 2, trobem 24 números quadràtics als quals es pot aplicar el mètode de Poincaré-Melnikov, i establim l'existència de quatre òrbites homoclíniques transversals per la gran majoria de valors del paràmetre de pertorbació. Estudiem també la continuació de les òrbites homoclíniques per a tots els valors del paràmetre de pertorbació en el cas del nombre d'argent sqrt(2) -1. En el cas d¿un tor amb bigotis tridimensional amb vector de freqüències donat per l'anomenat "nombre auri cúbic", establim l'existència de l¿escissió de separatrius exponencialment petita i detectem la transversalitat de 8 òrbites homoclíniques per la gran majoria de valors del paràmetre de pertorbació.

The goal of this thesis is the study of homoclinic orbits in conservative systems (area-preserving maps and Hamiltonian systems). We consider homoclinic (bi-asymptotic) orbits either to saddle periodic orbits or to whiskered tori. Such type orbits, called homoclinic by Poincaré, are of great interest in the theory of dynamical systems since their presence implies complicated dynamics. The thesis is divided in two parts according to two quite different topics considered. In the first part, we study area-preserving maps (APMs) with a nontransversal homoclinic orbit (homoclinic tangency) to a saddle fixed point in order to know the behavior of orbits near the given homoclinic trajectory. To this end, we construct first return maps, for which we use finitely-smooth normal forms of the saddle maps and introduce cross-coordinates. The fixed points of the first return maps correspond to single-round periodic orbits of the maps under consideration. Applying rescaling methods we derive the first return maps to the Hénon-like maps whose bifurcations are well known. Thus, translating the results obtained for the fixed points of the return maps to the periodic orbits, we prove the existence of cascades of elliptic periodic points. We also study the phenomenon of the coexistence of infinitely many single-round periodic orbits of different large periods (called global resonance). We consider the related problems in different types of APMs (symplectic maps and non-orientable APMs) with quadratic or cubic tangencies. We also establish the structure of 1:4 resonance for some conservative Hénon-like maps. The second part of the thesis is dedicated to the study of exponentially small splitting of separatrices arising from a perturbation of a Hamiltonian system with a homoclinic connection (separatrix). We consider a perturbation of an integrable Hamiltonian system having whiskered tori with coincident stable and unstable whiskers. Generally, in the perturbed system, the whiskers do not coincide anymore and our goal is to detect the transverse homoclinic orbits associated to the persistent whiskered tori. The perturbed system turns out to be not integrable due to the presence of these homoclinic trajectories and, consequently, there is chaotic dynamics near them. We give a suitable parametrization to the whiskers to determine the distance between them. This distance is given by the splitting function, and the simple zeros of this function give rise to transverse homoclinic orbits. We use the classical Poincaré-Melnikov approach to measure the splitting, although in the case of exponential smallness we have to ensure that the first order approximation overcome the error term. We consider Hamiltonian systems possessing two-dimensional whiskered tori with quadratic frequencies and three-dimensional whiskered tori with cubic golden frequency. In the two-dimensional case, we find 23 new quadratic numbers for which the Poincaré-Melnikov method can be applied and establish the existence of 4 transverse homoclinic orbits. We also study the continuation of the homoclinic orbits for all values of the parameter of perturbation in the case of the silver number sqrt(2)-1. For the three-dimensional whiskered torus with frequency vector given by the so-called "cubic golden number", we establish the existence of exponentially small splitting of separatrices and detect the transversality of 8 homoclinic orbits.

• DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES. APLICACIONES A

Delshams i Valdes, Amadeu
Participation in a competitive project

• Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Delshams i Valdes, Amadeu; Gidea, Marian; Roldan Gonzalez, Pablo
Discrete and continuous dynamical systems. Series A
Date of publication: 2012
Journal article

• On cascades of elliptic periodic points in area-preserving maps with homoclinic tangencies

Gonchenko, Marina; Delshams i Valdes, Amadeu
International Conference on Difference Equations and Applications
Presentation's date: 2012-07-27
Presentation of work at congresses

• Exponentially small splitting of separatrices for whiskered tori with quadratic frequencies

Gonchenko, Marina; Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere
Congreso No Lineal
Presentation's date: 2012-06-04
Presentation of work at congresses

• Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
Date: 2012-07
Report

The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton-Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel0nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example.

• Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Delshams i Valdes, Amadeu; Lazaro Ochoa, Jose Tomas; Gonchenko, Vladimir; Gonchenko ., S.V.; Sten'kin, Oleg
Date: 2012-01-12
Report

Abstract. 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.

• A geometric mechanism of diffusion: rigorous verification in a priori unstable Hamiltonian systems

Delshams i Valdes, Amadeu; Huguet, G.
Journal of differential equations
Date of publication: 2011-03-01
Journal article

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 a model, Mem. Amer. Math. Soc. 179 (844) (2006), viii + 141 pp.], and generalized in [A. Delshams, G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity 22 (8) (2009) 1997– 2077]. We provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform the straightforward computations along the proof and present the geometric mechanism of diffusion in an easily understandable way. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

• Jornades d'interacció entre sistemes dinàmics i equacions entre sistemes dinàmics i equacions en derivades parcials (JISD2011)

Martinez-seara Alonso, Maria Teresa; Delshams i Valdes, Amadeu; Cabre Vilagut, Xavier; Gonzalez Nogueras, Maria Del Mar
Participation in a competitive project

• Transversality of homoclinic orbits to hyberbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Pacha Andujar, Juan Ramon
Date: 2011
Report

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 approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and suffcient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop.

• Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Koltsova, O.; Pacha Andujar, Juan Ramon
Regular and chaotic dynamics
Date of publication: 2010-10
Journal article

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 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).

hyperbolic KAM tori - transverse homoclinic orbits - Melnikov method

• An accounting device for biasymptotic solutions: the scattering map in the restricted three body problem

Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
Date of publication: 2010
Book chapter

• A geometric mechanism of diffusion in a priori unstable Hamiltonian systems

Delshams i Valdes, Amadeu
International Symposium on Hamiltonian Systems and Celestial Mechanics
Presentation's date: 2010-12-02
Presentation of work at congresses

• The geometric mechanism of diffusion in mechanical systems

Delshams i Valdes, Amadeu; Huguet Casades, Gemma
Joint SIAM/RSME-SCM-SEMA Meeting
Presentation's date: 2010-05-31
Presentation of work at congresses

• Parabolic orbits in the restricted three-body problem

Delshams i Valdes, Amadeu; de la Rosa Ibarra, Abraham; Martinez-seara Alonso, Maria Teresa
International Symposium on Hamiltonian Systems and Celestial Mechanics
Presentation's date: 2010-12-01
Presentation of work at congresses

• Combining two scattering maps to obtain diffusion in the planar restricted planar 3-body problem

Delshams i Valdes, Amadeu; de la Rosa Ibarra, Abraham; Kaloshin, Vadim; Martinez-seara Alonso, Maria Teresa
International Symposium on Hamiltonian Systems and Celestial Mechanics
Presentation's date: 2010-11-29
Presentation of work at congresses

• DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES, ASTRODINÁMICA, NEUROCIENCIA Y OTRAS APLICACIONES

Blazquez Sanz, David; Olive Farre, Maria del Carme; Gomez-Ullate Oteiza, David; Morales Ruiz, Juan Jose; Canalias Vila, Elisabet; Martinez-seara Alonso, Maria Teresa; Baldoma Barraca, Inmaculada Concepcion; Fedorov, Yury; Garcia Taberner, Laura; Guardia Munarriz, Marcel; Guillamon Grabolosa, Antoni; Gonchenko, Marina; Gutiérrez Serrés, Pere; Lazaro Ochoa, Jose Tomas; Luque Jiménez, Alejandro; Martin De La Torre, Pablo; Masdemont Soler, Josep Joaquim; Olle Torner, Maria Mercedes; Pacha Andujar, Juan Ramon; Pantazi, Chara; Puig Sadurni, Joaquim; Ramirez Ros, Rafael; Roldan Gonzalez, Pablo; Villanueva Castelltort, Jordi; Tamarit, Anna; Olivé Farré, Carme; Morales Ruíz, Juan José; Larreal Barreto, Oswaldo José; Huguet Casades, Gemma; Gómez-Ullate Oteiza, David; Acosta Humanez, Primitivo Belen; Benita Bordes, Jose Manuel; Basak, Inna; Blázquez Sanz, David; de la Llave Canosa, Rafael; de la Rosa Ibarra, Abraham; Delshams i Valdes, Amadeu
Participation in a competitive project

• A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

Delshams i Valdes, Amadeu; Huguet Casades, Gemma
Date: 2010-07
Report

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. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

• A geometric mechanism for diffusion in a priori unstable hamiltonian systems

Delshams i Valdes, Amadeu
International Conference on Dynamical Systems. Celebrating Jacob Palis'70th birthday
Presentation's date: 2010
Presentation of work at congresses

• Euler's beta integral in Pietro Mengoli's works

Massa Esteve, Maria Rosa; Delshams i Valdes, Amadeu
Archive for history of exact sciences
Date of publication: 2009-03
Journal article

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.

• Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

Delshams i Valdes, Amadeu; Huguet Casades, Gemma
Nonlinearity
Date of publication: 2009
Journal article

• Arnold¿s mechanism of diffusion in the spatial circular Restricted Three Body Problem: a semi- numerical argument

Delshams i Valdes, Amadeu; Gidea, Marian; Roldan Gonzalez, Pablo
Dynamics, Topology and Computations
Presentation's date: 2009-06-03
Presentation of work at congresses

• Global instability in mechanical systems via geometrical methods

Delshams i Valdes, Amadeu
Congreso de Ecuaciones Diferenciales y Aplicaciones / Congreso de Matemática Aplicada
Presentation's date: 2009-09-25
Presentation of work at congresses

• SISTEMES DINAMICS DE LA UPC

Lazaro Ochoa, Jose Tomas; Olle Torner, Maria Mercedes; Gutiérrez Serrés, Pere; Huguet Casades, Gemma; Masdemont Soler, Josep Joaquim; Martinez-seara Alonso, Maria Teresa; Pantazi, Chara; Gomez-Ullate Oteiza, David; Luque Jiménez, Alejandro; Larreal Barreto, Oswaldo José; Morales Ruiz, Juan Jose; Pacha Andujar, Juan Ramon; Ramirez Ros, Rafael; Guillamon Grabolosa, Antoni; Martin De La Torre, Pablo; Canalias Vila, Elisabet; Puig Sadurni, Joaquim; Benita Bordes, Jose Manuel; Fedorov, Yury; Baldoma Barraca, Inmaculada Concepcion; Blazquez Sanz, David; de la Rosa Ibarra, Abraham; Guardia Munarriz, Marcel; Olive Farre, Maria del Carme; Acosta Humanez, Primitivo Belen; Gonchenko, Marina; Roldan Gonzalez, Pablo; de la Llave Canosa, Rafael; Villanueva Castelltort, Jordi; Delshams i Valdes, Amadeu
Participation in a competitive project

• Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method

Gutiérrez Serrés, Pere; Delshams i Valdes, Amadeu; Pacha Andujar, Juan Ramon
Date: 2009-12
Report

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 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)

• Computing the scattering map in the spatial hill's problem

Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
Discrete and continuous dynamical systems. Series B
Date of publication: 2008-09
Journal article

• Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
Advances in mathematics
Date of publication: 2008-02
Journal article

• Consideracions al voltant de la funció beta a la obra de Leonhard Euler (1707-1783)

Massa Esteve, Maria Rosa; Delshams i Valdes, Amadeu
Quaderns d'història de l'enginyeria
Date of publication: 2008-01
Journal article

• Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

Delshams i Valdes, Amadeu; de la Llave Canosa, Rafael; Martinez-seara Alonso, Maria Teresa
Date of publication: 2008-02
Book chapter

• The role of hyperbolic invariant objects: From Arnold diffusion to biological clocks

Huguet Casades, Gemma
Defense's date: 2008-10-16
School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
Theses

El marc d'aquesta tesi són els objectes invariants hiperbòlics (tors amb bigotis, cicles límit, NHIM,. . .), que constitueixen, per aquesta tesi, els objectes essencials per a l'estudi de diversos problemes des de la difusió d'Arnold fins als rellotges biològics. Treballem en tres temes diferents des d'un enfocament tant teòric com numèric, amb una especial atenció per a les aplicacions, especialment en neurobiologia:· Existència de difusió d'Arnold per a sistemes Hamiltonians a priori inestables· Algorismes numèrics ràpids per al càlcul de tors invariants i els "bigotis" associats, per a sistemes Hamiltonians utilitzant el mètode de la parametrització.· Càlcul d'isòcrones i corbes de resposta de fase (PRC) en sistemes neurobiològics usant el mètode de la parametrització.En la primera part de la tesi, hem considerat el cas d'un sistema Hamiltonià a priori inestable amb 2+1/2 graus de llibertat sotmès a una pertorbació de tipus general. "A priori inestable" significa que el sistema no pertorbat presenta un punt d'equilibri hiperbòlic amb una òrbita homoclínica associada. El resultat principal d'aquesta part de la tesi és que per a un conjunt genèric de pertorbacions prou regulars, el sistema presenta el fenòmen de la difusió d'Arnold, és a dir, existeixen trajectòries la variable acció de les quals experimenta un canvi d'ordre 1. La demostració es basa en un estudi detallat de les zones ressonants i els objectes invariants generats en elles, i ofereix una descripció completa de la geografia de les ressonàncies generades per una pertorbació genèrica.En la segona part d'aquest memòria, desenvolupem mètodes numèrics eficients que requereixen poca memòria i operacions per al càlcul de tors invariants i els "bigotis" associats en sistemes Hamiltonians (aplicacions simplèctiques i camps vectorials Hamiltonians).En particular, això inclou els objectes invariants involucrats en el mecanisme de la difusió d'Arnold, estudiat en el capítol anterior. Els algorismes es basen en el mètode de la parametrització i segueixen de prop demostracions recents del teorema KAM que no usen variables acció-angle. Donem detalls de la implementació numèrica que hem dut a terme i mostrem alguns exemples.En la darrera part de la tesi relacionem problemes de temps en sistemes biològics amb algunes eines conegudes de sistemes dinàmics. En particular, usem el mètode de la parametrització i les simetries de Lie per a calcular numèricament les isòcrones i les corbes de resposta de fase (PRC) associades a oscil·ladors i ho apliquem a diversos models biològics ben coneguts. A més a més, aconseguim estendre el càlcul de PRCs en un entorn de l'oscil·lador. Les PRCs són útils per a l'estudi de la sincronització d'oscil·ladors acoblats i una eina bàsica en biologia experimental (ritmes circadians, acoblament sinàptic i elèctric de neurones,. . . ).

• Contributiones a la sexta edicion del congreso Nolineal 2008

Marques Truyol, Francisco; Delshams i Valdes, Amadeu
Date of publication: 2008-06
Book

• Transversalidad de órbitas homoclínicas de puntos de equilibrio hiperbólicos de un hamiltoniano

Gutiérrez Serrés, Pere; Delshams i Valdes, Amadeu
NoLineal 2008
Presentation of work at congresses

• Diophantine conditions for cubic frequency vectors

Gonchenko, Marina; Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere
Conference on Stability and Instability in Mechanical Sysyems: Recent Progress and Mathematical Theory
Presentation of work at congresses

• Arnold difusion along nearly parabolic orbits for the planar restricted 3-body problem

Delshams i Valdes, Amadeu; Kaloshin, V; Martinez-seara Alonso, Maria Teresa
V International Symposium and Workshop on Hamiltonian Systems and Celestial Mechanics - HAMSYS-2008
Presentation of work at congresses

• Study of resonances for Diophantine cubic frequency vectors

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Date: 2008-09
Report

• Transversalidad de órbitas homoclínicas de puntos de equilibrio hiperbólicos de un hamiltoniano

Gutiérrez Serrés, Pere; Delshams i Valdes, Amadeu
Date: 2008-07
Report

• Diophantine conditions for cubic frequency vectors

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Date: 2008-06
Report

• Diophantine conditions for cubic frequency vectors

Delshams i Valdes, Amadeu; Gutiérrez Serrés, Pere; Gonchenko, Marina
Conference on Stability and Instability in Mechanical Sysyems: Recent Progress and Mathematical Theory
Presentation's date: 2008-09-25
Presentation of work at congresses

• Diophantine conditions for cubic frequency vectors

Delshams i Valdes, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
NoLineal 2008
Presentation of work at congresses

• An accounting device for biasymtotic solutions: the scattering map in the restricted three body problem

Delshams i Valdes, Amadeu; Masdemont Soler, Josep Joaquim; Roldan Gonzalez, Pablo
2nd Conference on Nonlinear Science and Complexity
Presentation of work at congresses

• Arnold's mechanism of diusion in the spatial circular Restricted Three Body Problem

Roldan Gonzalez, Pablo; Delshams i Valdes, Amadeu; Gidea, M
Conference on Bifurcations: Mathematical & Quantum Aspects & Applications, BIFUR 08
Presentation of work at congresses