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Exponentially Small Splitting of Separatrices Associated
to 3D Whiskered Tori with Cubic Frequencies
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
Comm. Math. Phys., 378(3):1931-1976, 2020
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An invitation to singular symplectic geometry
Roisin Braddell, Amadeu Delshams, Eva Miranda, Cédric Oms, Arnau Planas
Int. J. Geom. Methods Mod. Phys., 16:1940008, 16 pp, 2019
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Global Instability in the Restricted Planar Elliptic Three Body Problem
Amadeu Delshams, Vadim Kaloshin, Abraham de la Rosa, Tere M. Seara
Commun. Math. Phys., 366:1173–1228, 2019
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Quasi-periodic perturbations of heteroclinic attractor networks
Amadeu Delshams, Antoni Guillamon, Gemma Huguet
Chaos 28(10):103111, 19pp, 2018 (Featured article)
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Arnold diffusion for a complete family of perturbations with two independent harmonics
Amadeu Delshams, Rodrigo G. Schaefer
Discrete Contin. Dyn. Syst. 38(12):6047-6072, 2018
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Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems
Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng
Discrete Contin. Dyn. Syst. 38(12):i-iii, 2018
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Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro
Discrete Contin. Dyn. Syst. 38(9):4483-4507, 2018
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Shadowing of non-transversal heteroclinic chains
Amadeu Delshams, Adria Simon, Piotr Zgliczynski
J. Differential Equations 264:3619-3663, 2018
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Examples of integrable and non integrable systems on singular symplectic manifolds
Amadeu Delshams, Anna Kiesenhofer, Eva Miranda
J. Geom. Phys. 115:89-97, 2017
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Corrigendum: On dynamics and bifurcations of area-preserving maps with homoclinic tangencies
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko
Nonlinearity 30(3):C2, 2017.
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Arnold diffusion for a complete family of perturbations
Amadeu Delshams, Rodrigo G. Schaefer
Regul. Chaotic Dyn. 22(1):78-108, 2017
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On bifurcations of homoclinic tangencies in area-preserving maps on non-orientable manifolds.
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko
Difference equations,
discrete dynamical systems and applications,
edited by Ll. Alsedà, J.M. Cushing, S. Elaydi, A.A. Pinto,
pages 107-125, volume 180 of Springer Proc. Math. Stat., Springer, Berlin, 2016.
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Arnold's mechanism of diffusion in the
spatial circular restricted three-body problem: a semi-analytical argument
Amadeu Delshams, Marian Gidea, Pablo Roldán
Phys. D 334:29-48, 2016
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Exponentially Small Splitting of Separatrices and Transversality Associated to Whiskered Tori with Quadratic Frequency Ratio
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
SIAM J. Appl. Dyn. Syst. 15(2):981-1024, 2016.
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Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Adv. Math., 294:689-755, 2016
MR3479576
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A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
Extended abstracts Spring 2014—Hamiltonian systems and celestial mechanics, virus dynamics and evolution, edited by M. Corbera, J.M. Cors, J. Llibre, A. Korobeinikov, 31–37,
Trends Math. Res. Perspect. CRM Barc., 4, Birkhäuser/Springer, Cham, 2015.
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On dynamics and bifurcations of area-preserving maps with homoclinic tangencies
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko
Nonlinearity 28(9):3027-3071, 2015.
MR3403390
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On Bifurcations of Area-preserving and Nonorientable Maps
with Quadratic Homoclinic Tangencies
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko
Regul. Chaotic Dyn. 19(6):702-717, 2014.
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Continuation of the Exponentially Small Transversality
for the Splitting of Separatrices to a Whiskered Torus
with Silver Ratio
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
Regul. Chaotic Dyn. 19(6):663-680, 2014.
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Exponentially Small Lower Bounds for the Splitting of Separatrices to
Whiskered Tori with Frequencies of Constant Type
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24, No. 8 (2014) 1440011 (12 pages)
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Exponentially small asymptotic estimates for the splitting of separatrices
to whiskered tori with quadratic and cubic frequencies
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez
Electron. Res. Announc. Math. Sci. 21:41-61, 2014
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Abundance of attracting, repelling and elliptic
periodic orbits in two-dimensional reversible maps
A. Delshams, S.V. Gonchenko, V.S. Gonchenko, J.T. Lázaro,
O. Sten'kin
Nonlinearity 26:1-33, 2013
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Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian
system, via the Hamilton-Jacobi equation
Amadeu Delshams, Pere Gutiérrez, Juan R. Pacha
Physica D 243:64-85, 2013
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Transition map and shadowing lemma for normally hyperbolic invariant manifolds
Amadeu Delshams, Marian Gidea, Pablo Roldán
Discrete Contin. Dyn. Syst. 33(3):1089-1112, 2013
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A geometric mechanism of diffusion: Rigorous verification in
a priori unstable Hamiltonian systems
Amadeu Delshams, Gemma Huguet
J. Differential Equations, 250:2601-2623, 2011
MR2756078
MSN citations
Google Academic
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Transverse intersections between invariant manifolds of doubly
hyperbolic invariant tori, via the Poincaré-Mel'nikov method.
A. Delshams, P. Gutiérrez, O. Koltsova, J. R. Pacha
Regul. Chaotic Dyn., 15(2-3):222-236, 2010
MR 2011d:37100
MSN citations
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Geography of resonances and Arnold diffusion in
a priori unstable Hamiltonian systems
Amadeu Delshams, Gemma Huguet
Nonlinearity, 22:1997-2077, 2009
MR 2010h:37136
MSN citations
Google Academic
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Euler's beta integral in Pietro Mengoli's works
Ma Rosa Massa, Amadeu Delshams
Arch. Hist. Exact Sci., 3:325-356, 2009
MR 2501200
MSN citations
Google Academic
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Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation
Amadeu Delshams, Marian Gidea, Rafael de la Llave, Tere M. Seara
Hamiltonian dynamical systems and applications, edited by Walter Craig,
285-336,
NATO Sci. Peace Secur. Ser. B Phys. Biophys.,
Springer, Dordrecht, 2008
MR 2009k:37132
MSN citations
Google Academic
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Computing the scattering map in the spatial Hill's problem
Amadeu Delshams, Josep Masdemont, Pablo Roldán
Discrete Contin. Dyn. Syst. Ser. B, 10(2-3):455-483, 2008
MR 2425052
MSN citations
Google Academic
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Geometric properties of the scattering map of a normally
hyperbolic invariant manifold
Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Adv. Math., 217(3):1096-1153, 2008
MR 2008m:37100
MSN citations
Google Academic
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The scattering map in the planar Restricted Three Body Problem
E. Canalias, A. Delshams, J.J. Masdemont and P. Roldán
Celestial Mech. Dynam. Astronom., 95(1-4):155-171, 2006
MR 2007h:70021
MSN citations
Google Academic
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Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows
Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Adv. Math., 202(1):64-188, 2006
MR 2007a:37070
MSN citations
Google Academic
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Homoclinic Orbits for Invariant Tori
of Nearly Integrable Hamiltonian Systems
O. Yu. Koltsova, L. M. Lerman, A. Delshams, P. Gutiérrez
Dokl. Math., 73(2):217-220, 2006
MR 2008k:371236
MSN citations
Google Academic
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A Geometric Mechanism for Diffusion in Hamiltonian Systems
Overcoming the large Gap Problem: Heuristics and Rigorous
Verification on a Model
Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Mem. Amer. Math. Soc.,
179(844):1-141, 2006
MR 2184276
MSN citations
Google Academic
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Poincaré, creador de los métodos todavía modernos
en las ecuaciones diferenciales y en la mecánica celeste
Amadeu Delshams
Arbor: Ciencia, Pensamiento y Cultura,
CLXXVII(704):669-689, 2004
- Exponentially small splitting of separatrices for whiskered tori
in Hamiltonian systems
Amadeu Delshams, Pere Gutiérrez
J. Math. Sci., 128(2):2726-2746, 2005
Google Academic
-
Homoclinic orbits to invariant tori near a homoclinic orbit
to center-center-saddle equilibrium
Oksana Koltsova, Lev Lerman, Amadeu Delshams, Pere Gutiérrez
Phys. D., 201(3-4):268-290, 2005
MR 2005m:37140
MSN citations
Google Academic
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Pseudo-normal form near saddle-center or saddle-focus equilibria
Amadeu Delshams and J. Tomás Lázaro
J. Differential Equations, 208(2):312-343, 2005
MR 2005h:34101
MSN citations
Google Academic
-
Exponentially small splitting for whiskered tori in hamiltonian
systems: flow-box coordinates and upper bounds
Amadeu Delshams, Pere Gutiérrez, Tere M. Seara
Discrete Contin. Dyn. Syst., 11(4):785-826, 2004
MR 2005j:37102
MSN citations
Google Academic
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Exponentially small splitting for whiskered tori in hamiltonian
systems: continuation of transverse homoclinic orbits
Amadeu Delshams, Pere Gutiérrez
Discrete Contin. Dyn. Syst., 11(4):757-783, 2004
MR 2005j:37101
MSN citations
Google Academic
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Persistence of homoclinic orbits for billiards and twist maps
Sergey Bolotin, Amadeu Delshams, Rafael Ramírez-Ros
Nonlinearity, 17(4):1153-1177, 2004
MR 2005d:37126
MSN citations
Google Academic
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Mathematical modelling and simulation of cotransport phenomena
through flat sheet-supported liquid membranes
G. Benzal, A. Kumar, A. Delshams, A. M. Sastre
Hydrometallurgy, 74(1-2):117-30, 2004
MR 2024440
MSN citations
Google Academic
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A geometric mechanism for diffusion in Hamiltonian systems
overcoming the large gap problem: announcement of results
Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Electron. Res. Announc. Amer. Math. Soc.,
9:125-134, 2003
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Simulation of non-linear flow density in transition state from the
mathematical model of the diffusion of metal anions through a flat sheet
supported liquid membrane
G. Benzal, A. Kumar, A. M. Sastre, A. Delshams
J. Math. Chem., 34(1-2):1-27, 2003
- Exponentially small splitting of separatrices for whiskered tori
in Hamiltonian systems
Amadeu Delshams, Pere Gutiérrez
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI), 300:87-121, 2003
- A pseudo-normal form for planar vector fields
Amadeu Delshams, Antoni Guillamon, J. Tomás Lázaro
Qual. Theory Dyn. Syst., 3(1):51-82. 2002
- Homoclinic billiard orbits inside symmetrically perturbed
ellipsoids
Amadeu Delshams,
Yuri Fedorov,
Rafael Ramírez-Ros
Nonlinearity, 14:1141-1195, 2001
- Splitting potential and Poincaré-Melnikov method for whiskered
tori in Hamiltonian Systems
Amadeu Delshams,
Pere Gutiérrez
J. Nonlinear Sci., 10(4):433-476, 2000.
- KAM theory and a partial justification of Greene's criterion for
non-twist maps
Amadeu Delshams,
Rafael de la Llave
SIAM J. Math. Anal., 31(6):1235-1269, 2000.
- A geometric approach to the existence of orbits with unbounded energy
in generic periodic perturbations by a potential of generic geodesic
flows of T2
Amadeu Delshams,
Rafael de la Llave,
Tere M. Seara
Comm. Math. Phys., 209(2):353-392, 2000.
- Unbounded growth of energy in periodic perturbations of
geodesic flows of the torus
Amadeu Delshams,
Rafael de la Llave,
Tere M. Seara
J. Delgado, E.A. Lacomba, E. Pérez-Chavela, J. Llibre, editors,
Hamiltonian systems and Celestial mechanics (HAMSYS-98),
volume 6 of
World Scientific Monograph Series in Mathematics,
World Sci. Publishing, 2000.
Abstract (ASCII file),
PostScript (330 Kb),
zipped version (102 Kb).
- Splitting and Melnikov potentials in Hamiltonian systems
Amadeu Delshams,
Pere Gutiérrez
J. Delgado, E.A. Lacomba, E. Pérez-Chavela, J. Llibre, editors,
Hamiltonian systems and Celestial mechanics (HAMSYS-98),
volume 6 of
World Scientific Monograph Series in Mathematics,
World Sci. Publishing, 2000.
Abstract (ASCII file),
PostScript (277 Kb),
zipped version (106 Kb).
- Homoclinic orbits to invariant tori in Hamiltonian systems
Amadeu Delshams,
Pere Gutiérrez
C.K.R.T. Jones, A. Khibnik, editors,
Multiple-Time-Scale Dynamical Systems, pp. 1-28,
volume 122 of
IMA Vol. Math. Appl.,
Springer-Verlag, Berlin, 2000.
Abstract (ASCII file),
PostScript (311 Kb),
zipped version (119 Kb).
- Singular separatrix splitting and the Poincaré-Melnikov
method for area preserving maps
Amadeu Delshams,
Rafael Ramírez-Ros
B. Fiedler, K. Gröger, J. Sprekels, editors,
EQUADIFF 99, pp. 941-946,
World Sci. Publishing, 2000.
Abstract (ASCII file),
PostScript (153 Kb),
Pdf version (163 Kb),
zipped version (177 Kb).
- Singular separatrix splitting and Melnikov method:
An experimental study
Amadeu Delshams,
Rafael Ramírez-Ros
Experiment. Math., 8(1):29-48, 1999.
- Splitting of separatrices in Hamiltonian systems
and symplectic maps
Amadeu Delshams,
Tere M. Seara,
Rafael Ramírez-Ros
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 39-54. Kluwer, Dordrecht, 1999.
MR 2000j:37075
Abstract (ASCII file),
PostScript (262 Kb),
zipped version (102 Kb).
- Splitting of separatrices for (fast) quasiperiodic forcing
Amadeu Delshams,
Vassili Gelfreich,
Àngel Jorba,
Tere M. Seara
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 367-371. Kluwer, Dordrecht, 1999.
CMP 1720914
Abstract (ASCII file),
PostScript (108 Kb),
zipped version (42 Kb).
- Exponentially small estimates for KAM theorem
near an elliptic equilibrium point
Amadeu Delshams,
Pere Gutiérrez
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 386-390. Kluwer, Dordrecht, 1999.
MR 2000g:37087
Abstract (ASCII file),
PostScript (84 Kb),
zipped version (35 Kb).
- Effective stability in reversible systems
Amadeu Delshams,
José Tomás Lázaro
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 453-457. Kluwer, Dordrecht, 1999.
MR 2000g:37088
Abstract (ASCII file),
PostScript (96 Kb),
zipped version (39 Kb).
- Psi-series, singularities of solutions and integrability of
polynomial systems
Arnau Mir,
Amadeu Delshams
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 504-508. Kluwer, Dordrecht, 1999.
CMP 1720937
Abstract (ASCII file),
PostScript (71 Kb),
zipped version (30 Kb).
- Poincaré-Melnikov-Arnold method for twist maps
Amadeu Delshams,
Rafael Ramírez-Ros
C. Simó, editor,
Hamiltonian Systems with Three or More Degrees of Freedom,
volume 533 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
pp. 533-537. Kluwer, Dordrecht, 1999.
MR 2000g:37101
Abstract (ASCII file),
PostScript (125 Kb),
zipped version (54 Kb).
- Exponentially small splitting of separatrices for perturbed
integrable standard-like maps
Amadeu Delshams,
Rafael Ramírez-Ros
J. Nonlinear Sci., 8(3):317-352, 1998.
- Homoclinic orbits of twist maps and billiards
Amadeu Delshams,
Rafael Ramírez-Ros
D. Bambusi and G. Gaeta, editors,
Symmetry and Perturbation Theory, Quaderno GNFM, 54,
pp. 46-86. Firenze, Italy, 1998.
- Singular splitting of separatrices for the perturbed McMillan map
Amadeu Delshams,
Rafael Ramírez-Ros
XV Congress on Differential Equations and Applications/V Congress
on Applied Mathematics, Vol. I, II (Vigo, 1997),
pp. 243-248. Univ. Vigo, Vigo, 1998.
MR 99e:58156
Abstract (ASCII file),
PostScript (127 Kb),
zipped version (52 Kb).
- Homoclinic orbits of twist maps and billiards
Amadeu Delshams,
Rafael Ramírez-Ros
D. Bambusi and G. Gaeta, editors,
Symmetry and Perturbation Theory, Quaderno GNFM, 54,
pp. 46-86. Firenze, Italy, 1998.
Abstract (ASCII file),
PostScript (644 Kb),
zipped version (230 Kb).
- Melnikov potential for exact symplectic maps
Amadeu Delshams, Rafael Ramírez-Ros
Comm. Math. Phys., 190(1):213-245, 1997.
- Splitting of separatrices in Hamiltonian systems with one
and a half degrees of freedom
Amadeu Delshams, Tere M. Seara
Math. Phys. Electron. J., 3(1): 1-40, 1997.
- Exponentially small splitting of separatrices under fast
quasiperiodic forcing
Amadeu Delshams,
Vassili Gelfreich,
Àngel Jorba,
Tere M. Seara
Comm. Math. Phys., 189(1): 35-71, 1997.
- Psi-series of quadratic vector fields on the plane
Amadeu Delshams,
Arnau Mir
Publ. Mat., 41(1):101-125, 1997.
- Lower and upper bounds for the splitting of separatrices of the
pendulum under a fast quasiperiodic forcing
Amadeu Delshams,
Vassili Gelfreich,
Àngel Jorba,
Tere M. Seara
Electron. Res. Announc. Amer. Math. Soc.,
3:1-10, 1997.
- Estimates on invariant tori near an elliptic equilibrium point
of a Hamiltonian system
Amadeu Delshams,
Pere Gutiérrez
J. Differential Equations, 131:277-303, 1996.
- Effective stability and KAM theory
Amadeu Delshams,
Pere Gutiérrez
J. Differential Equations, 128:415-490, 1996.
- Poincaré-Melnikov-Arnold method for analytic planar maps
Amadeu Delshams,
Rafael Ramírez-Ros
Nonlinearity, 9:1-26, 1996.
- On Birkhoff's conjecture about convex billiards
Amadeu Delshams,
Rafael Ramírez-Ros
M. Sofonea and J.-N. Corvellec, editors,
Proceedings of the 2nd Catalan Days on Applied Mathematics
(Odeillo, 1995), Collect. Études,
pp. 85-94.
Presses Univ. de Perpignan, Perpignan, 1995.
- An asymptotic expression for the splitting of separatrices of the
rapidly forced pendulum
Amadeu Delshams,
Tere M. Seara
Comm. Math. Phys., 150(3): 433-463, 1992.
- Effective stability for a Hamiltonian system near an elliptic equilibrium point,
with an application to the restricted three-body problem
Antonio Giorgilli, Amadeu Delshams, Ernest Fontich, Luigi Galgani, Carles Simó
J. Differential Equations, 77(1):167-198, 1989.