December  2018, 38(12): 6215-6239. doi: 10.3934/dcds.2018267

Some questions looking for answers in dynamical systems

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain

Dedicated to my friend, professor Rafael de la Llave Canosa, for his 60th birthday

Received  November 2017 Revised  July 2018 Published  September 2018

Dynamical systems appear in many models in all sciences and in technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms.

Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results.

In these notes a sample of questions, for which the author is not aware of the existence of a good solution, are presented. Of course, it is easy to largely extend the list.

Citation: Carles Simó. Some questions looking for answers in dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6215-6239. doi: 10.3934/dcds.2018267
References:
[1]

G. Arioli and H. Koch, The critical renormalization fixed point for commuting pairs of area-preserving maps, Comm. Math. Phys., 295 (2010), 415-429.  doi: 10.1007/s00220-009-0922-1.  Google Scholar

[2]

V. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40.   Google Scholar

[3]

V. Arnold, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.   Google Scholar

[4]

V. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

V. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967.  Google Scholar

[6]

V. Arnold, V. Kozlov and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics. Dynamical systems. III, 3$^{rd}$ edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations, 197 (2004), 45-72.  doi: 10.1016/j.jde.2003.07.005.  Google Scholar

[8]

I. BaldomáE. FontichR. de la Llave and P. Martín, The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic points, Discrete Contin. Dyn. Syst. Ser. A, 17 (2007), 835-865.  doi: 10.3934/dcds.2007.17.835.  Google Scholar

[9]

I. BaldomáE. Fontich and P. Martín, Gevrey estimates for one-dimensional parabolic invariant manifolds of non-hyperbolic fixed points, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4159-4190.  doi: 10.3934/dcds.2017177.  Google Scholar

[10]

I. Baldomá and À. Haro, One dimensional invariant manifolds of Gevrey type in real-analytic maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 295-322.  doi: 10.3934/dcdsb.2008.10.295.  Google Scholar

[11]

L. Biasco and L. Chierchia, On the measure of Lagrangian invariant tori in nearly-integrable mechanical system, Rend. Lincei Mat. Appl., 26 (2015), 423-432.  doi: 10.4171/RLM/713.  Google Scholar

[12]

K. Bjerklöv, SNA's in the quasiperiodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[13]

A. BounemouraB. Fayad and L. Niederman, Super exponential stability of quasiperiodic motion in Hamiltonian systems, Comm. Math. Phys., 350 (2017), 361-386.  doi: 10.1007/s00220-016-2782-9.  Google Scholar

[14]

H. Broer and C. Simó, Hill's equation with quasiperiodic forcing: Resonance tongues, instability pockets and global phenomena, Bul. Soc. Bras. Mat., 29 (1998), 253-293.  doi: 10.1007/BF01237651.  Google Scholar

[15]

H. BroerC. Simó and R. Vitolo, Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance "bubble", Physica D, 237 (2008), 1773-1799.  doi: 10.1016/j.physd.2008.01.026.  Google Scholar

[16]

H. BroerC. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnold resonance web, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 769-787.   Google Scholar

[17]

H. BroerC. Simó and R. Vitolo, Chaos and quasiperiodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905.  doi: 10.3934/dcdsb.2010.14.871.  Google Scholar

[18]

R. Calleja and R. de la Llave, A numerically accessible criterion for the breakdown of quasiperiodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058.  doi: 10.1088/0951-7715/23/9/001.  Google Scholar

[19]

R. Calleja and J-Ll. Figueras, Collision of invariant bundles of quasiperiodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10pp. doi: 10.1063/1.4737205.  Google Scholar

[20]

M. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1977-2026.  doi: 10.1088/0951-7715/25/7/1997.  Google Scholar

[21]

D. del-Castillo-NegreteJ. Greene and P. Morrison, Area preserving nontwist maps: Periodic orbits and transition to chaos, Physica D, 91 (1996), 1-23.  doi: 10.1016/0167-2789(95)00257-X.  Google Scholar

[22]

D. del-Castillo-NegreteJ. Greene and P. Morrison, Renormalization and transition to chaos in area preserving nontwist maps, Physica D, 100 (1997), 311-329.  doi: 10.1016/S0167-2789(96)00200-X.  Google Scholar

[23]

C. Cheng and Y. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 275-292.  doi: 10.1007/BF00053456.  Google Scholar

[24]

C. Cheng and Y. Sun, Existence of periodically invariant curves in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 293-303.  doi: 10.1007/BF00053457.  Google Scholar

[25]

B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979), 264-379.  doi: 10.1016/0370-1573(79)90023-1.  Google Scholar

[26]

A. DelshamsM. Gonchenko and P. Gutiérrez, Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies, Electronic Research Announcements in Math. Sci., 21 (2014), 41-61.  doi: 10.3934/era.2014.21.41.  Google Scholar

[27]

A. Delshams and P. Gutiérrez, Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system, J. Differential Equations, 131 (1996), 277-303.  doi: 10.1006/jdeq.1996.0165.  Google Scholar

[28]

A. Delshams and R. de la Llave, KAM theory and a partial justification of Greene's criterion for non-twist maps, SIAM J. of Math. Anal., 31 (2000), 1235-1269.  doi: 10.1137/S003614109834908X.  Google Scholar

[29]

A. Delshams and R. Schaefer, Arnold diffusion for a complete family of perturbations, Regular and Chaotic Dynamics, 22 (2017), 78-108.  doi: 10.1134/S1560354717010051.  Google Scholar

[30]

A. Delshams and R. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 6047-6072.   Google Scholar

[31]

H. Dullin and J. Meiss, Resonances and twist in volume-preserving maps, SIAM J. Appl. Dyn. Syst., 11 (2012), 319-349.  doi: 10.1137/110846865.  Google Scholar

[32]

F. DumortierS. IbáñezH. Kokubu and C. Simó, About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 4435-4471.  doi: 10.3934/dcds.2013.33.4435.  Google Scholar

[33]

C. Falcolini and R. de la Llave, A rigorous partial justification of Greene criterion, J. Statist. Phys., 67 (1992), 609-643.  doi: 10.1007/BF01049722.  Google Scholar

[34]

J.-L. Figueras and À. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16pp. doi: 10.1063/1.4938185.  Google Scholar

[35]

J.-L. Figueras and À. Haro, A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1095-1107.  doi: 10.3934/dcdss.2016043.  Google Scholar

[36]

J-Ll. FiguerasÀ. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[37]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Th. and Dynam. Syst., 10 (1990), 319-346.  doi: 10.1017/S0143385700005575.  Google Scholar

[38]

E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergod. Th. and Dynam. Syst., 10 (1990), 295-318.  doi: 10.1017/S0143385700005563.  Google Scholar

[39]

E. Fontich, C. Simó and A. Vieiro, Splitting of the separatrices after a Hamiltonian-Hopf bifurcation under periodic forcing, preprint, 2018. Google Scholar

[40]

E. Fontich, C. Simó and A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, preprint, 2018. Google Scholar

[41]

A. Fox and J. Meiss, Critical invariant circles in asymmetric and multiharmonic generalized standard maps, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 1004-1026.  doi: 10.1016/j.cnsns.2013.07.028.  Google Scholar

[42]

L. Garrido and C. Simó, Some ideas about strange attractors, in Proceed. Sitges Conference on Dynamical Systems and Chaos, Lect. Notes in Physics, 179, 1–28, Springer, 1983.  Google Scholar

[43]

V. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201 (1999), 155-216.  doi: 10.1007/s002200050553.  Google Scholar

[44]

V. GelfreichV. LazutkinC. Simó and M. Tabanov, Fern-like structures in the wild set of the standard and semistandard maps in $\mathbb{C}×\mathbb{C}$, Internat. J. of Bif. and Chaos, 2 (1992), 353-370.  doi: 10.1142/S0218127492000343.  Google Scholar

[45]

V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511-536.  doi: 10.3934/dcdsb.2008.10.511.  Google Scholar

[46]

A. GiorgilliA. DelshamsE. FontichL. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[47]

S. GonchenkoI. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. of Bif. and Chaos, 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[48]

A. González, À. Haro and R. de la Llave, Singularity theory for non-twist KAM tori, Mem. Amer. Math. Soc., 227 (2014), vi+115 pp.  Google Scholar

[49]

A. González, À. Haro and R. de la Llave, Geometric and computational aspects of singularity theory for KAM tori, work in progress. Google Scholar

[50]

C. GrebogiE. OttS. Pelikan and J. Yorke, Strange attractors that are not chaotic, Physica D, 13 (1984), 261-268.  doi: 10.1016/0167-2789(84)90282-3.  Google Scholar

[51]

J. Greene, A method for determining stochastic transition, J. Math. Phys., 6 (1979), 1183-1201.  doi: 10.2172/6191337.  Google Scholar

[52]

M. GuardiaP. Martín and T. Martínez Seara, Oscillatory motions for the restricted planar circular three-body problem, Invent. math., 203 (2016), 417-492.  doi: 10.1007/s00222-015-0591-y.  Google Scholar

[53]

À. Haro, The Primitive Function of an Exact Symplectomorphism, Ph.D. thesis, Departament de Matemática Aplicada i Anàlisi, Universitat de Barcelona, 1998. doi: 10.1088/0951-7715/13/5/304.  Google Scholar

[54]

À. Haro, M. Canadell, J-Ll. Figueras, A. Luque and J-M. Mondelo, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Applied Mathematical Sciences, 195, Springer, 2016. doi: 10.1007/978-3-319-29662-3.  Google Scholar

[55]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: rigorous results, J. Differential Equations, 228 (2006), 530-579.  doi: 10.1016/j.jde.2005.10.005.  Google Scholar

[56]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.  doi: 10.3934/dcdsb.2006.6.1261.  Google Scholar

[57]

À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp. doi: 10.1063/1.2150947.  Google Scholar

[58]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.  doi: 10.1137/050637327.  Google Scholar

[59]

À. Haro and J. Puig, Strange non-chaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7pp. doi: 10.1063/1.2259821.  Google Scholar

[60]

J. Howard and S. Hohs, Stochasticity and reconnection in Hamiltonian systems, Physical Review A, 29 (1984), 418-421.  doi: 10.1103/PhysRevA.29.418.  Google Scholar

[61]

M. Irigoyen and C. Simó, Non-integrability of the $J_2$ problem, Celestial Mech. Dynam. Astronom., 55 (1993), 281-287.  doi: 10.1007/BF00692515.  Google Scholar

[62]

T. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), vi+106 pp. doi: 10.1090/memo/0945.  Google Scholar

[63]

T. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289.  doi: 10.1007/s00220-009-0753-0.  Google Scholar

[64]

À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124.  doi: 10.1016/0022-0396(92)90107-X.  Google Scholar

[65]

À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. of Math. Anal., 27 (1996), 1704-1737.  doi: 10.1137/S0036141094276913.  Google Scholar

[66]

À. Jorba and J.-C. Tatjer, A mechanism for the fractalization of invariant curves in quasiperiodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[67]

À. JorbaJ-C. TatjerC. Núñez and R. Obaya, Old and new results on strange non-chaotic attractors, Internat. J. of Bif. and Chaos, 17 (2007), 3895-3928.  doi: 10.1142/S0218127407019780.  Google Scholar

[68]

K. Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115.  doi: 10.1143/PTP.71.1112.  Google Scholar

[69]

G. Keller, A note on strange non-chaotic attractors, Fund. Math., 151 (1996), 139-148.   Google Scholar

[70]

J. Ketoja and R. MacKay, Fractal boundary for the existence of invariant circles for area-preserving maps: Observations and renormalisation explanation, Physica D, 35 (1989), 318-334.  doi: 10.1016/0167-2789(89)90073-0.  Google Scholar

[71]

K. KhaninJ. Lopes-Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows, Nonlinearity, 19 (2006), 2727-2753.  doi: 10.1088/0951-7715/19/12/001.  Google Scholar

[72]

K. KhaninJ. Lopes-Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231.  doi: 10.1007/s00220-006-0125-y.  Google Scholar

[73]

A. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.- London, 1964.  Google Scholar

[74]

H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst. Ser. A, 11 (2004), 881-909.  doi: 10.3934/dcds.2004.11.881.  Google Scholar

[75]

A. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation on the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

[76]

E. Lacomba and C. Simó, Analysis of quadruple collision in some degenerate cases, Celestial Mech. Dynam. Astronom. 28 (1982), 49–62. doi: 10.1007/BF01230659.  Google Scholar

[77]

E. Lacomba and C. Simó, Regularization of simultaneous binary collisions, J. Differential Equations 98 (1992), 241–259. doi: 10.1016/0022-0396(92)90092-2.  Google Scholar

[78]

V. Lazutkin and C. Simó, Homoclinic orbits in the complex domain, Internat. J. of Bif. and Chaos, 7 (1997), 253-274.  doi: 10.1142/S0218127497000182.  Google Scholar

[79]

A. Litvak-Hinenzon and V. Rom-Kedar, Symmetry-breaking perturbations and strange attractors, Physical Review E, 55 (1997), 4964-4978.  doi: 10.1103/PhysRevE.55.4964.  Google Scholar

[80]

R. de la Llave, A tutorial on KAM theory, in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69. 175–292. Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/pspum/069/1858536.  Google Scholar

[81]

R. de la LlaveA. GonzálezÀ. Jorba and J. Villanueva, KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895.  doi: 10.1088/0951-7715/18/2/020.  Google Scholar

[82]

R. de la Llave and A. Olvera, The obstruction criterion for non existence of invariant circles and renormalization, Nonlinearity, 19 (2006), 1907-1937.  doi: 10.1088/0951-7715/19/8/008.  Google Scholar

[83]

J. Llibre, R. Moeckel and Simó, Central Configurations, Periodic Orbits and Hamiltonian Systems, in Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, 240 pp., 2015. Google Scholar

[84]

H. Lomelí and R. Calleja, Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map, Chaos, 16 (2006), 023117, 8pp. doi: 10.1063/1.2179647.  Google Scholar

[85]

R. MacKay, A renormalization approach to invariant circles in area-preserving maps, Physica D, 7 (1983), 283-300.  doi: 10.1016/0167-2789(83)90131-8.  Google Scholar

[86]

R. MacKay, Hyperbolic Cantori have dimension zero, J. Phys. A: Math. Gen., 20 (1987), L559-L561.  doi: 10.1088/0305-4470/20/9/002.  Google Scholar

[87]

R. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187.  doi: 10.1088/0951-7715/5/1/007.  Google Scholar

[88]

R. MacKay, Renormalization in Area-Preserving Maps, Advanced Series in Nonlinear Dynamics 6, World Scientific, 1993. doi: 10.1142/9789814354462.  Google Scholar

[89]

R. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002.  Google Scholar

[90]

R. Martínez and C. Simó, Simultaneous binary collisions in the planar four body problem, Nonlinearity, 12 (1999), 903-930.  doi: 10.1088/0951-7715/12/4/310.  Google Scholar

[91]

R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems, Nonlinearity, 13 (2000), 2107-2130.  doi: 10.1088/0951-7715/13/6/312.  Google Scholar

[92]

R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348.  doi: 10.1134/S1560354709030010.  Google Scholar

[93]

R. Martínez and C. Simó, Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 1-24.  doi: 10.3934/dcds.2011.29.1.  Google Scholar

[94]

R. Martínez and C. Simó, The invariant manifolds at infinity of the restricted three-body problem and the boundaries of bounded motion, Regular and Chaotic Dynamics, 19 (2014), 745-765.  doi: 10.1134/S1560354714060112.  Google Scholar

[95]

R. Martínez and C. Simó, Return maps, dynamical consequences and applications, Qualitative Theory of Dynamical Systems, 14 (2015), 353-379.  doi: 10.1007/s12346-015-0154-z.  Google Scholar

[96]

R. Martínez, C. Simó and A. Susín, Regularizable and non-regularizable collisions in N-body problems, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 12–19.  Google Scholar

[97]

J. Mather, Non existence of invariant circles, Ergod. Th. and Dynam. Syst., 4 (1984), 301-309.  doi: 10.1017/S0143385700002455.  Google Scholar

[98]

J. Mather, A criterion for the non existence of invariant circles, IHES Publ. Math., 63 (1986), 153-204.   Google Scholar

[99]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88.  doi: 10.1016/0022-0396(73)90077-6.  Google Scholar

[100]

A. MedvedevA. Neishtadt and D. Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.  Google Scholar

[101]

J. Meiss, N. Miguel, C. Simó and A. Vieiro, Stickiness, accelerator modes and anomalous diffusion due to a resonance bubble emerging from a Hopf-saddle-node bifurcation in 3D volume preserving maps, preprint, 2018. Google Scholar

[102]

N. MiguelC. Simó and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics, 18 (2013), 469-489.  doi: 10.1134/S1560354713050018.  Google Scholar

[103]

J.-J. Morales, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics 179, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[104]

J.-J. Morales and J.-M. Peris, On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse Math., 8 (1999), 125-141.  doi: 10.5802/afst.925.  Google Scholar

[105]

J.-J. Morales and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods and Applications of Analysis, 8 (2001), 33-95.  doi: 10.4310/MAA.2001.v8.n1.a3.  Google Scholar

[106]

J.-J. Morales and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems Ⅱ, Methods and Applications of Analysis, 8 (2001), 97-111.  doi: 10.4310/MAA.2001.v8.n1.a3.  Google Scholar

[107]

J.-J. Morales and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods and Applications of Analysis, 8 (2001), 113-120.  doi: 10.4310/MAA.2001.v8.n1.a5.  Google Scholar

[108]

J.-J. MoralesJ.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Annales Sci. de l'ENS 4e série, 40 (2007), 845-884.  doi: 10.1016/j.ansens.2007.09.002.  Google Scholar

[109]

J.-J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations, 129 (1996), 111-135.  doi: 10.1006/jdeq.1996.0113.  Google Scholar

[110]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617.  doi: 10.1007/BF02180145.  Google Scholar

[111]

P. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Physics of Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.  Google Scholar

[112]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[113]

J. Moser, Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Princeton Univ. Press, 1973.  Google Scholar

[114]

A. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika, 48 (1984), 133-139.  doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[115]

N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys, 32 (1977), 5-66.   Google Scholar

[116]

A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica D, 26 (1987), 181-192.  doi: 10.1016/0167-2789(87)90222-3.  Google Scholar

[117]

G. Piftankin and D. Treschev, Separatrix maps in Hamiltonian systems, Russian Mathematical Surveys, 62 (2007), 219-322.  doi: 10.1070/RM2007v062n02ABEH004396.  Google Scholar

[118]

J. Puig and C. Simó, Analytic families of reducible linear quasiperiodic differential equations, Ergod. Th. and Dynam. Syst., 26 (2006), 481-524.  doi: 10.1017/S0143385705000362.  Google Scholar

[119]

J. Puig and C. Simó, Resonance tongues and Spectral gaps in quasiperiodic Schrödinger operators with one or more frequencies. A numerical exploration, J. of Dynamics and Diff. Eq., 23 (2011), 649-669.  doi: 10.1007/s10884-010-9199-5.  Google Scholar

[120]

J. Puig and C. Simó, Resonance tongues in the quasiperiodic Hill-Schr¨odinger equation with three frequencies, Regular and Chaotic Dynamics, 16 (2011), 61-78.  doi: 10.1134/S1560354710520047.  Google Scholar

[121]

H. Rüssmann and I. Kleine Nenner, Über invariante kurven differenzierbarer abbildungen eines kreisringes, Nach. Akad. Wiss. Göttingen Math.-Phys.Kl. II, 1970 (1970), 67-105.   Google Scholar

[122]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Geometric Dynamics (Rio de Janeiro, 1981), 677–718, Lecture Notes in Math. 1007, Springer, 1983. doi: 10.1007/BFb0061441.  Google Scholar

[123]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regular and Chaotic Chaotic Dynamics, 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

[124]

C. Simó, Estimates of the error in normal forms of Hamiltonian systems. Applications to effective stability. Examples, in Long-term Dynamical Behavior of Natural and Artificial N-body Systems, A.E. Roy, ed., 481–503, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 246, Kluwer Acad. Publ., Dordrecht, 1988.  Google Scholar

[125]

C. Simó, Analytical and numerical computation of invariant manifolds, in Modern Methods in Celestial Mechanics, D. Benest and C. Froeschlé, editors, 285–330, Editions Frontières, 1990. Google Scholar

[126]

C. Simó, Measuring the lack of integrability in the J2 problem, in Predictability, Stability and Chaos in the N-body Dynamical Systems, A.E. Roy, editor, 305–309, NATO Adv. Sci. Inst. Ser. B Phys., 272, Plenum, New York, 1991.  Google Scholar

[127]

C. Simó, Averaging under fast quasiperiodic forcing, Hamiltonian Mechanics, 13–34, NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, 1994.  Google Scholar

[128]

C. Simó, Invariant curves of perturbations of non-twist integrable area preserving maps, Regular and Chaotic Dynamics, 3 (1998), 180-195.  doi: 10.1070/rd1998v003n03ABEH000088.  Google Scholar

[129]

C. Simó, Effective computations in celestial mechanics and astrodynamics, in Modern Methods of Analytical Mechanics and their Applications, V. V. Rumyantsev and A. V. Karapetyan, ed., CISM Courses and Lectures 387, 55–102, Springer, 1998. Google Scholar

[130]

C. Simó, Measuring the total amount of chaos in some Hamiltonian systems, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 5135-5164.  doi: 10.3934/dcds.2014.34.5135.  Google Scholar

[131]

C. Simó, Experiments looking for theoretical predictions, Indagationes Mathematicae, 27 (2016), 1069-1080.  doi: 10.1016/j.indag.2015.11.013.  Google Scholar

[132]

C. Simó, P. Sousa-Silva and M. Terra, Practical stability domains near L4, 5 in the restricted three-body problem: Some preliminary facts, in Progress and Challenges in Dynamical Systems, 54, 367–382, Springer, 2013. doi: 10.1007/978-3-642-38830-9_23.  Google Scholar

[133]

C. Simó and T. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Physica D, 140 (2000), 1-32.  doi: 10.1016/S0167-2789(99)00211-0.  Google Scholar

[134]

C. Simó and A. Susín, Connections between invariant manifolds in the collision manifold of the planar three-body problem, in The Geometry of Hamiltonian Systems, 497–518, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991. doi: 10.1007/978-1-4613-9725-0_18.  Google Scholar

[135]

C. Simó and A. Susín, Equilibrium connections in the triple collision manifold, in Predictability, Stability and Chaos in the N-Body Dynamical Systems, 481–491, NATO Adv. Sci. Inst. Ser. B Phys., 272, Plenum, New York, 1991.  Google Scholar

[136]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics Doklady, 5 (1960), 647-650.   Google Scholar

[137]

Y. Sun and L. Zhou, Stickiness in three-dimensional volume preserving mappings, Celestial Mech. Dynam. Astronom., 103 (2009), 119-131.  doi: 10.1007/s10569-008-9173-2.  Google Scholar

[138]

V. Szebehely, Theory of Orbits, Academic Press, 1967. Google Scholar

[139]

S. Tompaidis, Approximation of invariant surfaces by periodic orbits in high-dimensional maps: some rigorous results, Experiment. Math., 5 (1996), 197-209.  doi: 10.1080/10586458.1996.10504588.  Google Scholar

[140]

G. Zaslavskii and B. Chirikov, Stochastic instability of nonlinear oscillations, Soviet Physics Uspekhi, 14 (1972), 549-568.  doi: 10.3367/UFNr.0105.197109a.0003.  Google Scholar

[141]

S. Ziglin, Branching of solutions and non existence of first integrals in Hamiltonian mechanics I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96.   Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, The critical renormalization fixed point for commuting pairs of area-preserving maps, Comm. Math. Phys., 295 (2010), 415-429.  doi: 10.1007/s00220-009-0922-1.  Google Scholar

[2]

V. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40.   Google Scholar

[3]

V. Arnold, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.   Google Scholar

[4]

V. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

V. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967.  Google Scholar

[6]

V. Arnold, V. Kozlov and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics. Dynamical systems. III, 3$^{rd}$ edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.  Google Scholar

[7]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations, 197 (2004), 45-72.  doi: 10.1016/j.jde.2003.07.005.  Google Scholar

[8]

I. BaldomáE. FontichR. de la Llave and P. Martín, The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic points, Discrete Contin. Dyn. Syst. Ser. A, 17 (2007), 835-865.  doi: 10.3934/dcds.2007.17.835.  Google Scholar

[9]

I. BaldomáE. Fontich and P. Martín, Gevrey estimates for one-dimensional parabolic invariant manifolds of non-hyperbolic fixed points, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4159-4190.  doi: 10.3934/dcds.2017177.  Google Scholar

[10]

I. Baldomá and À. Haro, One dimensional invariant manifolds of Gevrey type in real-analytic maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 295-322.  doi: 10.3934/dcdsb.2008.10.295.  Google Scholar

[11]

L. Biasco and L. Chierchia, On the measure of Lagrangian invariant tori in nearly-integrable mechanical system, Rend. Lincei Mat. Appl., 26 (2015), 423-432.  doi: 10.4171/RLM/713.  Google Scholar

[12]

K. Bjerklöv, SNA's in the quasiperiodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[13]

A. BounemouraB. Fayad and L. Niederman, Super exponential stability of quasiperiodic motion in Hamiltonian systems, Comm. Math. Phys., 350 (2017), 361-386.  doi: 10.1007/s00220-016-2782-9.  Google Scholar

[14]

H. Broer and C. Simó, Hill's equation with quasiperiodic forcing: Resonance tongues, instability pockets and global phenomena, Bul. Soc. Bras. Mat., 29 (1998), 253-293.  doi: 10.1007/BF01237651.  Google Scholar

[15]

H. BroerC. Simó and R. Vitolo, Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance "bubble", Physica D, 237 (2008), 1773-1799.  doi: 10.1016/j.physd.2008.01.026.  Google Scholar

[16]

H. BroerC. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnold resonance web, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 769-787.   Google Scholar

[17]

H. BroerC. Simó and R. Vitolo, Chaos and quasiperiodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905.  doi: 10.3934/dcdsb.2010.14.871.  Google Scholar

[18]

R. Calleja and R. de la Llave, A numerically accessible criterion for the breakdown of quasiperiodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058.  doi: 10.1088/0951-7715/23/9/001.  Google Scholar

[19]

R. Calleja and J-Ll. Figueras, Collision of invariant bundles of quasiperiodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10pp. doi: 10.1063/1.4737205.  Google Scholar

[20]

M. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1977-2026.  doi: 10.1088/0951-7715/25/7/1997.  Google Scholar

[21]

D. del-Castillo-NegreteJ. Greene and P. Morrison, Area preserving nontwist maps: Periodic orbits and transition to chaos, Physica D, 91 (1996), 1-23.  doi: 10.1016/0167-2789(95)00257-X.  Google Scholar

[22]

D. del-Castillo-NegreteJ. Greene and P. Morrison, Renormalization and transition to chaos in area preserving nontwist maps, Physica D, 100 (1997), 311-329.  doi: 10.1016/S0167-2789(96)00200-X.  Google Scholar

[23]

C. Cheng and Y. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 275-292.  doi: 10.1007/BF00053456.  Google Scholar

[24]

C. Cheng and Y. Sun, Existence of periodically invariant curves in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 293-303.  doi: 10.1007/BF00053457.  Google Scholar

[25]

B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979), 264-379.  doi: 10.1016/0370-1573(79)90023-1.  Google Scholar

[26]

A. DelshamsM. Gonchenko and P. Gutiérrez, Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies, Electronic Research Announcements in Math. Sci., 21 (2014), 41-61.  doi: 10.3934/era.2014.21.41.  Google Scholar

[27]

A. Delshams and P. Gutiérrez, Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system, J. Differential Equations, 131 (1996), 277-303.  doi: 10.1006/jdeq.1996.0165.  Google Scholar

[28]

A. Delshams and R. de la Llave, KAM theory and a partial justification of Greene's criterion for non-twist maps, SIAM J. of Math. Anal., 31 (2000), 1235-1269.  doi: 10.1137/S003614109834908X.  Google Scholar

[29]

A. Delshams and R. Schaefer, Arnold diffusion for a complete family of perturbations, Regular and Chaotic Dynamics, 22 (2017), 78-108.  doi: 10.1134/S1560354717010051.  Google Scholar

[30]

A. Delshams and R. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 6047-6072.   Google Scholar

[31]

H. Dullin and J. Meiss, Resonances and twist in volume-preserving maps, SIAM J. Appl. Dyn. Syst., 11 (2012), 319-349.  doi: 10.1137/110846865.  Google Scholar

[32]

F. DumortierS. IbáñezH. Kokubu and C. Simó, About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 4435-4471.  doi: 10.3934/dcds.2013.33.4435.  Google Scholar

[33]

C. Falcolini and R. de la Llave, A rigorous partial justification of Greene criterion, J. Statist. Phys., 67 (1992), 609-643.  doi: 10.1007/BF01049722.  Google Scholar

[34]

J.-L. Figueras and À. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16pp. doi: 10.1063/1.4938185.  Google Scholar

[35]

J.-L. Figueras and À. Haro, A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1095-1107.  doi: 10.3934/dcdss.2016043.  Google Scholar

[36]

J-Ll. FiguerasÀ. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[37]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Th. and Dynam. Syst., 10 (1990), 319-346.  doi: 10.1017/S0143385700005575.  Google Scholar

[38]

E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergod. Th. and Dynam. Syst., 10 (1990), 295-318.  doi: 10.1017/S0143385700005563.  Google Scholar

[39]

E. Fontich, C. Simó and A. Vieiro, Splitting of the separatrices after a Hamiltonian-Hopf bifurcation under periodic forcing, preprint, 2018. Google Scholar

[40]

E. Fontich, C. Simó and A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, preprint, 2018. Google Scholar

[41]

A. Fox and J. Meiss, Critical invariant circles in asymmetric and multiharmonic generalized standard maps, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 1004-1026.  doi: 10.1016/j.cnsns.2013.07.028.  Google Scholar

[42]

L. Garrido and C. Simó, Some ideas about strange attractors, in Proceed. Sitges Conference on Dynamical Systems and Chaos, Lect. Notes in Physics, 179, 1–28, Springer, 1983.  Google Scholar

[43]

V. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201 (1999), 155-216.  doi: 10.1007/s002200050553.  Google Scholar

[44]

V. GelfreichV. LazutkinC. Simó and M. Tabanov, Fern-like structures in the wild set of the standard and semistandard maps in $\mathbb{C}×\mathbb{C}$, Internat. J. of Bif. and Chaos, 2 (1992), 353-370.  doi: 10.1142/S0218127492000343.  Google Scholar

[45]

V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511-536.  doi: 10.3934/dcdsb.2008.10.511.  Google Scholar

[46]

A. GiorgilliA. DelshamsE. FontichL. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[47]

S. GonchenkoI. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. of Bif. and Chaos, 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[48]

A. González, À. Haro and R. de la Llave, Singularity theory for non-twist KAM tori, Mem. Amer. Math. Soc., 227 (2014), vi+115 pp.  Google Scholar

[49]

A. González, À. Haro and R. de la Llave, Geometric and computational aspects of singularity theory for KAM tori, work in progress. Google Scholar

[50]

C. GrebogiE. OttS. Pelikan and J. Yorke, Strange attractors that are not chaotic, Physica D, 13 (1984), 261-268.  doi: 10.1016/0167-2789(84)90282-3.  Google Scholar

[51]

J. Greene, A method for determining stochastic transition, J. Math. Phys., 6 (1979), 1183-1201.  doi: 10.2172/6191337.  Google Scholar

[52]

M. GuardiaP. Martín and T. Martínez Seara, Oscillatory motions for the restricted planar circular three-body problem, Invent. math., 203 (2016), 417-492.  doi: 10.1007/s00222-015-0591-y.  Google Scholar

[53]

À. Haro, The Primitive Function of an Exact Symplectomorphism, Ph.D. thesis, Departament de Matemática Aplicada i Anàlisi, Universitat de Barcelona, 1998. doi: 10.1088/0951-7715/13/5/304.  Google Scholar

[54]

À. Haro, M. Canadell, J-Ll. Figueras, A. Luque and J-M. Mondelo, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Applied Mathematical Sciences, 195, Springer, 2016. doi: 10.1007/978-3-319-29662-3.  Google Scholar

[55]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: rigorous results, J. Differential Equations, 228 (2006), 530-579.  doi: 10.1016/j.jde.2005.10.005.  Google Scholar

[56]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.  doi: 10.3934/dcdsb.2006.6.1261.  Google Scholar

[57]

À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp. doi: 10.1063/1.2150947.  Google Scholar

[58]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.  doi: 10.1137/050637327.  Google Scholar

[59]

À. Haro and J. Puig, Strange non-chaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7pp. doi: 10.1063/1.2259821.  Google Scholar

[60]

J. Howard and S. Hohs, Stochasticity and reconnection in Hamiltonian systems, Physical Review A, 29 (1984), 418-421.  doi: 10.1103/PhysRevA.29.418.  Google Scholar

[61]

M. Irigoyen and C. Simó, Non-integrability of the $J_2$ problem, Celestial Mech. Dynam. Astronom., 55 (1993), 281-287.  doi: 10.1007/BF00692515.  Google Scholar

[62]

T. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), vi+106 pp. doi: 10.1090/memo/0945.  Google Scholar

[63]

T. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289.  doi: 10.1007/s00220-009-0753-0.  Google Scholar

[64]

À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124.  doi: 10.1016/0022-0396(92)90107-X.  Google Scholar

[65]

À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. of Math. Anal., 27 (1996), 1704-1737.  doi: 10.1137/S0036141094276913.  Google Scholar

[66]

À. Jorba and J.-C. Tatjer, A mechanism for the fractalization of invariant curves in quasiperiodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[67]

À. JorbaJ-C. TatjerC. Núñez and R. Obaya, Old and new results on strange non-chaotic attractors, Internat. J. of Bif. and Chaos, 17 (2007), 3895-3928.  doi: 10.1142/S0218127407019780.  Google Scholar

[68]

K. Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115.  doi: 10.1143/PTP.71.1112.  Google Scholar

[69]

G. Keller, A note on strange non-chaotic attractors, Fund. Math., 151 (1996), 139-148.   Google Scholar

[70]

J. Ketoja and R. MacKay, Fractal boundary for the existence of invariant circles for area-preserving maps: Observations and renormalisation explanation, Physica D, 35 (1989), 318-334.  doi: 10.1016/0167-2789(89)90073-0.  Google Scholar

[71]

K. KhaninJ. Lopes-Dias and J. Marklof, Renormalization of multidimensional Hamiltonian flows, Nonlinearity, 19 (2006), 2727-2753.  doi: 10.1088/0951-7715/19/12/001.  Google Scholar

[72]

K. KhaninJ. Lopes-Dias and J. Marklof, Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231.  doi: 10.1007/s00220-006-0125-y.  Google Scholar

[73]

A. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.- London, 1964.  Google Scholar

[74]

H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst. Ser. A, 11 (2004), 881-909.  doi: 10.3934/dcds.2004.11.881.  Google Scholar

[75]

A. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation on the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

[76]

E. Lacomba and C. Simó, Analysis of quadruple collision in some degenerate cases, Celestial Mech. Dynam. Astronom. 28 (1982), 49–62. doi: 10.1007/BF01230659.  Google Scholar

[77]

E. Lacomba and C. Simó, Regularization of simultaneous binary collisions, J. Differential Equations 98 (1992), 241–259. doi: 10.1016/0022-0396(92)90092-2.  Google Scholar

[78]

V. Lazutkin and C. Simó, Homoclinic orbits in the complex domain, Internat. J. of Bif. and Chaos, 7 (1997), 253-274.  doi: 10.1142/S0218127497000182.  Google Scholar

[79]

A. Litvak-Hinenzon and V. Rom-Kedar, Symmetry-breaking perturbations and strange attractors, Physical Review E, 55 (1997), 4964-4978.  doi: 10.1103/PhysRevE.55.4964.  Google Scholar

[80]

R. de la Llave, A tutorial on KAM theory, in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69. 175–292. Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/pspum/069/1858536.  Google Scholar

[81]

R. de la LlaveA. GonzálezÀ. Jorba and J. Villanueva, KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895.  doi: 10.1088/0951-7715/18/2/020.  Google Scholar

[82]

R. de la Llave and A. Olvera, The obstruction criterion for non existence of invariant circles and renormalization, Nonlinearity, 19 (2006), 1907-1937.  doi: 10.1088/0951-7715/19/8/008.  Google Scholar

[83]

J. Llibre, R. Moeckel and Simó, Central Configurations, Periodic Orbits and Hamiltonian Systems, in Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, 240 pp., 2015. Google Scholar

[84]

H. Lomelí and R. Calleja, Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map, Chaos, 16 (2006), 023117, 8pp. doi: 10.1063/1.2179647.  Google Scholar

[85]

R. MacKay, A renormalization approach to invariant circles in area-preserving maps, Physica D, 7 (1983), 283-300.  doi: 10.1016/0167-2789(83)90131-8.  Google Scholar

[86]

R. MacKay, Hyperbolic Cantori have dimension zero, J. Phys. A: Math. Gen., 20 (1987), L559-L561.  doi: 10.1088/0305-4470/20/9/002.  Google Scholar

[87]

R. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187.  doi: 10.1088/0951-7715/5/1/007.  Google Scholar

[88]

R. MacKay, Renormalization in Area-Preserving Maps, Advanced Series in Nonlinear Dynamics 6, World Scientific, 1993. doi: 10.1142/9789814354462.  Google Scholar

[89]

R. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002.  Google Scholar

[90]

R. Martínez and C. Simó, Simultaneous binary collisions in the planar four body problem, Nonlinearity, 12 (1999), 903-930.  doi: 10.1088/0951-7715/12/4/310.  Google Scholar

[91]

R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems, Nonlinearity, 13 (2000), 2107-2130.  doi: 10.1088/0951-7715/13/6/312.  Google Scholar

[92]

R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348.  doi: 10.1134/S1560354709030010.  Google Scholar

[93]

R. Martínez and C. Simó, Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 1-24.  doi: 10.3934/dcds.2011.29.1.  Google Scholar

[94]

R. Martínez and C. Simó, The invariant manifolds at infinity of the restricted three-body problem and the boundaries of bounded motion, Regular and Chaotic Dynamics, 19 (2014), 745-765.  doi: 10.1134/S1560354714060112.  Google Scholar

[95]

R. Martínez and C. Simó, Return maps, dynamical consequences and applications, Qualitative Theory of Dynamical Systems, 14 (2015), 353-379.  doi: 10.1007/s12346-015-0154-z.  Google Scholar

[96]

R. Martínez, C. Simó and A. Susín, Regularizable and non-regularizable collisions in N-body problems, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 12–19.  Google Scholar

[97]

J. Mather, Non existence of invariant circles, Ergod. Th. and Dynam. Syst., 4 (1984), 301-309.  doi: 10.1017/S0143385700002455.  Google Scholar

[98]

J. Mather, A criterion for the non existence of invariant circles, IHES Publ. Math., 63 (1986), 153-204.   Google Scholar

[99]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88.  doi: 10.1016/0022-0396(73)90077-6.  Google Scholar

[100]

A. MedvedevA. Neishtadt and D. Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.  Google Scholar

[101]

J. Meiss, N. Miguel, C. Simó and A. Vieiro, Stickiness, accelerator modes and anomalous diffusion due to a resonance bubble emerging from a Hopf-saddle-node bifurcation in 3D volume preserving maps, preprint, 2018. Google Scholar

[102]

N. MiguelC. Simó and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics, 18 (2013), 469-489.  doi: 10.1134/S1560354713050018.  Google Scholar

[103]

J.-J. Morales, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics 179, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[104]

J.-J. Morales and J.-M. Peris, On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse Math., 8 (1999), 125-141.  doi: 10.5802/afst.925.  Google Scholar

[105]

J.-J. Morales and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods and Applications of Analysis, 8 (2001), 33-95.  doi: 10.4310/MAA.2001.v8.n1.a3.  Google Scholar

[106]

J.-J. Morales and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems Ⅱ, Methods and Applications of Analysis, 8 (2001), 97-111.  doi: 10.4310/MAA.2001.v8.n1.a3.  Google Scholar

[107]

J.-J. Morales and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods and Applications of Analysis, 8 (2001), 113-120.  doi: 10.4310/MAA.2001.v8.n1.a5.  Google Scholar

[108]

J.-J. MoralesJ.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Annales Sci. de l'ENS 4e série, 40 (2007), 845-884.  doi: 10.1016/j.ansens.2007.09.002.  Google Scholar

[109]

J.-J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations, 129 (1996), 111-135.  doi: 10.1006/jdeq.1996.0113.  Google Scholar

[110]

A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617.  doi: 10.1007/BF02180145.  Google Scholar

[111]

P. Morrison, Magnetic field lines, Hamiltonian dynamics, and nontwist systems, Physics of Plasmas, 7 (2000), 2279-2289.  doi: 10.1063/1.874062.  Google Scholar

[112]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[113]

J. Moser, Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Princeton Univ. Press, 1973.  Google Scholar

[114]

A. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika, 48 (1984), 133-139.  doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[115]

N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys, 32 (1977), 5-66.   Google Scholar

[116]

A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica D, 26 (1987), 181-192.  doi: 10.1016/0167-2789(87)90222-3.  Google Scholar

[117]

G. Piftankin and D. Treschev, Separatrix maps in Hamiltonian systems, Russian Mathematical Surveys, 62 (2007), 219-322.  doi: 10.1070/RM2007v062n02ABEH004396.  Google Scholar

[118]

J. Puig and C. Simó, Analytic families of reducible linear quasiperiodic differential equations, Ergod. Th. and Dynam. Syst., 26 (2006), 481-524.  doi: 10.1017/S0143385705000362.  Google Scholar

[119]

J. Puig and C. Simó, Resonance tongues and Spectral gaps in quasiperiodic Schrödinger operators with one or more frequencies. A numerical exploration, J. of Dynamics and Diff. Eq., 23 (2011), 649-669.  doi: 10.1007/s10884-010-9199-5.  Google Scholar

[120]

J. Puig and C. Simó, Resonance tongues in the quasiperiodic Hill-Schr¨odinger equation with three frequencies, Regular and Chaotic Dynamics, 16 (2011), 61-78.  doi: 10.1134/S1560354710520047.  Google Scholar

[121]

H. Rüssmann and I. Kleine Nenner, Über invariante kurven differenzierbarer abbildungen eines kreisringes, Nach. Akad. Wiss. Göttingen Math.-Phys.Kl. II, 1970 (1970), 67-105.   Google Scholar

[122]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Geometric Dynamics (Rio de Janeiro, 1981), 677–718, Lecture Notes in Math. 1007, Springer, 1983. doi: 10.1007/BFb0061441.  Google Scholar

[123]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regular and Chaotic Chaotic Dynamics, 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

[124]

C. Simó, Estimates of the error in normal forms of Hamiltonian systems. Applications to effective stability. Examples, in Long-term Dynamical Behavior of Natural and Artificial N-body Systems, A.E. Roy, ed., 481–503, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 246, Kluwer Acad. Publ., Dordrecht, 1988.  Google Scholar

[125]

C. Simó, Analytical and numerical computation of invariant manifolds, in Modern Methods in Celestial Mechanics, D. Benest and C. Froeschlé, editors, 285–330, Editions Frontières, 1990. Google Scholar

[126]

C. Simó, Measuring the lack of integrability in the J2 problem, in Predictability, Stability and Chaos in the N-body Dynamical Systems, A.E. Roy, editor, 305–309, NATO Adv. Sci. Inst. Ser. B Phys., 272, Plenum, New York, 1991.  Google Scholar

[127]

C. Simó, Averaging under fast quasiperiodic forcing, Hamiltonian Mechanics, 13–34, NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, 1994.  Google Scholar

[128]

C. Simó, Invariant curves of perturbations of non-twist integrable area preserving maps, Regular and Chaotic Dynamics, 3 (1998), 180-195.  doi: 10.1070/rd1998v003n03ABEH000088.  Google Scholar

[129]

C. Simó, Effective computations in celestial mechanics and astrodynamics, in Modern Methods of Analytical Mechanics and their Applications, V. V. Rumyantsev and A. V. Karapetyan, ed., CISM Courses and Lectures 387, 55–102, Springer, 1998. Google Scholar

[130]

C. Simó, Measuring the total amount of chaos in some Hamiltonian systems, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 5135-5164.  doi: 10.3934/dcds.2014.34.5135.  Google Scholar

[131]

C. Simó, Experiments looking for theoretical predictions, Indagationes Mathematicae, 27 (2016), 1069-1080.  doi: 10.1016/j.indag.2015.11.013.  Google Scholar

[132]

C. Simó, P. Sousa-Silva and M. Terra, Practical stability domains near L4, 5 in the restricted three-body problem: Some preliminary facts, in Progress and Challenges in Dynamical Systems, 54, 367–382, Springer, 2013. doi: 10.1007/978-3-642-38830-9_23.  Google Scholar

[133]

C. Simó and T. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Physica D, 140 (2000), 1-32.  doi: 10.1016/S0167-2789(99)00211-0.  Google Scholar

[134]

C. Simó and A. Susín, Connections between invariant manifolds in the collision manifold of the planar three-body problem, in The Geometry of Hamiltonian Systems, 497–518, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991. doi: 10.1007/978-1-4613-9725-0_18.  Google Scholar

[135]

C. Simó and A. Susín, Equilibrium connections in the triple collision manifold, in Predictability, Stability and Chaos in the N-Body Dynamical Systems, 481–491, NATO Adv. Sci. Inst. Ser. B Phys., 272, Plenum, New York, 1991.  Google Scholar

[136]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics Doklady, 5 (1960), 647-650.   Google Scholar

[137]

Y. Sun and L. Zhou, Stickiness in three-dimensional volume preserving mappings, Celestial Mech. Dynam. Astronom., 103 (2009), 119-131.  doi: 10.1007/s10569-008-9173-2.  Google Scholar

[138]

V. Szebehely, Theory of Orbits, Academic Press, 1967. Google Scholar

[139]

S. Tompaidis, Approximation of invariant surfaces by periodic orbits in high-dimensional maps: some rigorous results, Experiment. Math., 5 (1996), 197-209.  doi: 10.1080/10586458.1996.10504588.  Google Scholar

[140]

G. Zaslavskii and B. Chirikov, Stochastic instability of nonlinear oscillations, Soviet Physics Uspekhi, 14 (1972), 549-568.  doi: 10.3367/UFNr.0105.197109a.0003.  Google Scholar

[141]

S. Ziglin, Branching of solutions and non existence of first integrals in Hamiltonian mechanics I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96.   Google Scholar

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