December  2020, 40(12): 6795-6813. doi: 10.3934/dcds.2020166

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

1. 

Department of Mathematical Sciences, Yeshiva University, New York, NY 10016, USA

2. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

3. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, 08028, Spain

* Corresponding author

Received  December 2019 Published  December 2020 Early access  March 2020

Fund Project: M.G. was partially supported by NSF grant DMS-1814543.
R.L. was partially supported by NSF grant DMS-1800241.
T.S. was partially supported by the MINECO-FEDER Grant MTM2015-65715-P and PGC2018-098676-B-100, the Catalan Grant 2017SGR1049, and the ICREA Academia 2019 award.
Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester.

We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions – which can be verified in concrete systems by finite calculations – are satisfied.

In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikov-type integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the $ C^r $-topology, for $ r \in [3, \infty) \cup \{\omega\} $. Our method does not require that the unperturbed Hamiltonian system is convex, or that the perturbation is polynomial, which are non-generic properties.

Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.

Citation: Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
References:
[1]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom, in Vladimir I. Arnold - Collected Works, Collected Works, 1, Springer, Berlin, Heidelberg, 423–427. doi: 10.1007/978-3-642-01742-1_26.

[2]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.

[3]

P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.

[4]

P. BernardV. Kaloshin and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders, Acta Math., 217 (2016), 1-79.  doi: 10.1007/s11511-016-0141-5.

[5]

M. Berti and P. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 395-450.  doi: 10.1016/S0294-1449(01)00084-1.

[6]

M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl. (9), 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.

[7]

S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063.  doi: 10.1088/0951-7715/19/9/003.

[8]

M. J. Capinski and M. Gidea, Arnold diffusion, quantitative estimates and stochastic behavior in the three-body problem, preprint, arXiv: 1812.03665.

[9]

M. CapinskiM. Gidea and R. De la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: Mechanism and numerical verification, Nonlinearity, 30 (2017), 329-360.  doi: 10.1088/1361-6544/30/1/329.

[10]

M. J. Capiński and P. Zgliczyński, Transition tori in the planar restricted elliptic three-body problem, Nonlinearity, 24 (2011), 1395-1432.  doi: 10.1088/0951-7715/24/5/002.

[11]

M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order: I. The equation $y''- k (1-y^2) y'+ y = b\lambda k \cos(\lambda t+ \alpha)$, $k$ large, J. London Math. Soc., 20 (1945), 180-189.  doi: 10.1112/jlms/s1-20.3.180.

[12]

Q. Chen and R. de la Llave, Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems, (2019).

[13]

C.-Q. Cheng, Variational methods for the problem of Arnold diffusion, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,337–365. doi: 10.1007/978-1-4020-6964-2_14.

[14]

C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.  doi: 10.4310/jdg/1246888485.

[15]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp.

[16]

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

[17]

W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.

[18]

A. DelshamsR. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2, Comm. Math. Phys., 209 (2000), 353-392.  doi: 10.1007/PL00020961.

[19]

A. DelshamsR. de la Llave and T. M Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math., 202 (2006), 64-188.  doi: 10.1016/j.aim.2005.03.005.

[20]

A. DelshamsR. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153.  doi: 10.1016/j.aim.2007.08.014.

[21]

A. DelshamsR. de la Llave and T. M. Seara, Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion, Adv. Math., 294 (2016), 689-755.  doi: 10.1016/j.aim.2015.11.010.

[22]

A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,285–336. doi: 10.1007/978-1-4020-6964-2_13.

[23]

A. DelshamsM. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Phys. D, 334 (2016), 29-48.  doi: 10.1016/j.physd.2016.06.005.

[24]

A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077.  doi: 10.1088/0951-7715/22/8/013.

[25]

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

[26]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst., 38 (2018), 6047-6072.  doi: 10.3934/dcds.2018261.

[27]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[28]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.

[29]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346.  doi: 10.1017/S0143385700005575.

[30]

V. Gelfreich and D. Turaev, Unbounded energy growth in Hamiltonian systems with a slowly varying parameter, Comm. Math. Phys., 283 (2008), 769-794.  doi: 10.1007/s00220-008-0518-1.

[31]

V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547.  doi: 10.1007/s00220-017-2867-0.

[32]

M. Gidea and R. de la Llave, Perturbations of geodesic flows by recurrent dynamics, J. Eur. Math. Soc. (JEMS), 19 (2017), 905-956.  doi: 10.4171/JEMS/683.

[33]

M. Gidea and R. de la Llave, Global Melnikov theory in Hamiltonian systems with general time-dependent perturbations, J. Nonlinear Sci., 28 (2018), 1657-1707.  doi: 10.1007/s00332-018-9461-2.

[34]

M. Gidea, R. de la Llave and M. Musser, Global effect of non-conservative perturbations on homoclinic orbits, preprint, arXiv: 1909.02080.

[35]

M. Gidea, R. de la Llave and T. M-Seara, Accessibility and control in nearly integrable Hamiltonian systems, 2019.

[36]

M. GideaR. de la Llave and T. M-Seara, A general mechanism of diffusion in Hamiltonian systems: Qualitative results, Comm. Pure Appl. Math., 73 (2020), 150-209.  doi: 10.1002/cpa.21856.

[37]

M. Gidea and J.-P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314.

[38]

A. Granados, Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators, Phys. D, 351/352 (2017), 14-29.  doi: 10.1016/j.physd.2017.04.003.

[39]

M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323.

[40]

M. GuzzoE. Lega and C. Froeschlé, A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems, Phys. D, 238 (2009), 1797-1807.  doi: 10.1016/j.physd.2009.06.009.

[41]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[42]

V. Kaloshin and K. Zhang, Arnold diffusion for smooth convex systems of two and a half degrees of freedom, Nonlinearity, 28 (2015), 2699-2720.  doi: 10.1088/0951-7715/28/8/2699.

[43] A. Liapounoff, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies, 17, Princeton University Press, Princeton, NJ; Oxford University Press, London, 1947. 
[44]

A. Luque and D. Peralta-Salas, Arnold diffusion of charged particles in ABC magnetic fields, J. Nonlinear Sci., 27 (2017), 721-774.  doi: 10.1007/s00332-016-9349-y.

[45]

J. N. Mather, Arnold diffusion. I. Announcement of results, J. Math. Sci. (N.Y.), 124 (2004), 5275-5289.  doi: 10.1023/B:JOTH.0000047353.78307.09.

[46]

J. N. Mather, Arnold diffusion by variational methods, in Essays in Mathematics and Its Applications, Springer, Heidelberg, 2012,271–285. doi: 10.1007/978-3-642-28821-0_11.

[47]

R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial Mechanics, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002,163–171. doi: 10.1090/conm/292/04922.

[48]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.

[49]

G. N. Piftankin, Diffusion speed in the Mather problem, Dokl. Akad. Nauk, 408 (2006), 736-737.  doi: 10.1088/0951-7715/19/11/007.

[50]

H. Poincaré, Les Methodes Nouvelles de la Mecanique Celeste. Tome III, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987.

[51]

P. K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknechta, 2 (1938), 83-94. 

[52]

C. Robinson, Differentiable conjugacy near compact invariant manifolds, Bol. Soc. Brasil. Mat., 2 (1971), 33-44.  doi: 10.1007/BF02584805.

[53] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, NJ, 1965. 
[54]

D. Treschev, Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems, Nonlinearity, 15 (2002), 2033-2052.  doi: 10.1088/0951-7715/15/6/313.

[55]

D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841.  doi: 10.1088/0951-7715/17/5/014.

[56]

D. Treschev, Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems, Nonlinearity, 25 (2012), 2717-2757.  doi: 10.1088/0951-7715/25/9/2717.

[57]

J. Villanueva, Kolmogorov theorem revisited, J. Differential Equations, 244 (2008), 2251-2276.  doi: 10.1016/j.jde.2008.02.010.

show all references

References:
[1]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom, in Vladimir I. Arnold - Collected Works, Collected Works, 1, Springer, Berlin, Heidelberg, 423–427. doi: 10.1007/978-3-642-01742-1_26.

[2]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.

[3]

P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.

[4]

P. BernardV. Kaloshin and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders, Acta Math., 217 (2016), 1-79.  doi: 10.1007/s11511-016-0141-5.

[5]

M. Berti and P. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 395-450.  doi: 10.1016/S0294-1449(01)00084-1.

[6]

M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl. (9), 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.

[7]

S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063.  doi: 10.1088/0951-7715/19/9/003.

[8]

M. J. Capinski and M. Gidea, Arnold diffusion, quantitative estimates and stochastic behavior in the three-body problem, preprint, arXiv: 1812.03665.

[9]

M. CapinskiM. Gidea and R. De la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: Mechanism and numerical verification, Nonlinearity, 30 (2017), 329-360.  doi: 10.1088/1361-6544/30/1/329.

[10]

M. J. Capiński and P. Zgliczyński, Transition tori in the planar restricted elliptic three-body problem, Nonlinearity, 24 (2011), 1395-1432.  doi: 10.1088/0951-7715/24/5/002.

[11]

M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order: I. The equation $y''- k (1-y^2) y'+ y = b\lambda k \cos(\lambda t+ \alpha)$, $k$ large, J. London Math. Soc., 20 (1945), 180-189.  doi: 10.1112/jlms/s1-20.3.180.

[12]

Q. Chen and R. de la Llave, Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems, (2019).

[13]

C.-Q. Cheng, Variational methods for the problem of Arnold diffusion, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,337–365. doi: 10.1007/978-1-4020-6964-2_14.

[14]

C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.  doi: 10.4310/jdg/1246888485.

[15]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp.

[16]

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

[17]

W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.

[18]

A. DelshamsR. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2, Comm. Math. Phys., 209 (2000), 353-392.  doi: 10.1007/PL00020961.

[19]

A. DelshamsR. de la Llave and T. M Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math., 202 (2006), 64-188.  doi: 10.1016/j.aim.2005.03.005.

[20]

A. DelshamsR. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153.  doi: 10.1016/j.aim.2007.08.014.

[21]

A. DelshamsR. de la Llave and T. M. Seara, Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion, Adv. Math., 294 (2016), 689-755.  doi: 10.1016/j.aim.2015.11.010.

[22]

A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,285–336. doi: 10.1007/978-1-4020-6964-2_13.

[23]

A. DelshamsM. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Phys. D, 334 (2016), 29-48.  doi: 10.1016/j.physd.2016.06.005.

[24]

A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077.  doi: 10.1088/0951-7715/22/8/013.

[25]

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

[26]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst., 38 (2018), 6047-6072.  doi: 10.3934/dcds.2018261.

[27]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226.  doi: 10.1512/iumj.1972.21.21017.

[28]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.

[29]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346.  doi: 10.1017/S0143385700005575.

[30]

V. Gelfreich and D. Turaev, Unbounded energy growth in Hamiltonian systems with a slowly varying parameter, Comm. Math. Phys., 283 (2008), 769-794.  doi: 10.1007/s00220-008-0518-1.

[31]

V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547.  doi: 10.1007/s00220-017-2867-0.

[32]

M. Gidea and R. de la Llave, Perturbations of geodesic flows by recurrent dynamics, J. Eur. Math. Soc. (JEMS), 19 (2017), 905-956.  doi: 10.4171/JEMS/683.

[33]

M. Gidea and R. de la Llave, Global Melnikov theory in Hamiltonian systems with general time-dependent perturbations, J. Nonlinear Sci., 28 (2018), 1657-1707.  doi: 10.1007/s00332-018-9461-2.

[34]

M. Gidea, R. de la Llave and M. Musser, Global effect of non-conservative perturbations on homoclinic orbits, preprint, arXiv: 1909.02080.

[35]

M. Gidea, R. de la Llave and T. M-Seara, Accessibility and control in nearly integrable Hamiltonian systems, 2019.

[36]

M. GideaR. de la Llave and T. M-Seara, A general mechanism of diffusion in Hamiltonian systems: Qualitative results, Comm. Pure Appl. Math., 73 (2020), 150-209.  doi: 10.1002/cpa.21856.

[37]

M. Gidea and J.-P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314.

[38]

A. Granados, Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators, Phys. D, 351/352 (2017), 14-29.  doi: 10.1016/j.physd.2017.04.003.

[39]

M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323.

[40]

M. GuzzoE. Lega and C. Froeschlé, A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems, Phys. D, 238 (2009), 1797-1807.  doi: 10.1016/j.physd.2009.06.009.

[41]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.

[42]

V. Kaloshin and K. Zhang, Arnold diffusion for smooth convex systems of two and a half degrees of freedom, Nonlinearity, 28 (2015), 2699-2720.  doi: 10.1088/0951-7715/28/8/2699.

[43] A. Liapounoff, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies, 17, Princeton University Press, Princeton, NJ; Oxford University Press, London, 1947. 
[44]

A. Luque and D. Peralta-Salas, Arnold diffusion of charged particles in ABC magnetic fields, J. Nonlinear Sci., 27 (2017), 721-774.  doi: 10.1007/s00332-016-9349-y.

[45]

J. N. Mather, Arnold diffusion. I. Announcement of results, J. Math. Sci. (N.Y.), 124 (2004), 5275-5289.  doi: 10.1023/B:JOTH.0000047353.78307.09.

[46]

J. N. Mather, Arnold diffusion by variational methods, in Essays in Mathematics and Its Applications, Springer, Heidelberg, 2012,271–285. doi: 10.1007/978-3-642-28821-0_11.

[47]

R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial Mechanics, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002,163–171. doi: 10.1090/conm/292/04922.

[48]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.

[49]

G. N. Piftankin, Diffusion speed in the Mather problem, Dokl. Akad. Nauk, 408 (2006), 736-737.  doi: 10.1088/0951-7715/19/11/007.

[50]

H. Poincaré, Les Methodes Nouvelles de la Mecanique Celeste. Tome III, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987.

[51]

P. K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknechta, 2 (1938), 83-94. 

[52]

C. Robinson, Differentiable conjugacy near compact invariant manifolds, Bol. Soc. Brasil. Mat., 2 (1971), 33-44.  doi: 10.1007/BF02584805.

[53] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, NJ, 1965. 
[54]

D. Treschev, Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems, Nonlinearity, 15 (2002), 2033-2052.  doi: 10.1088/0951-7715/15/6/313.

[55]

D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841.  doi: 10.1088/0951-7715/17/5/014.

[56]

D. Treschev, Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems, Nonlinearity, 25 (2012), 2717-2757.  doi: 10.1088/0951-7715/25/9/2717.

[57]

J. Villanueva, Kolmogorov theorem revisited, J. Differential Equations, 244 (2008), 2251-2276.  doi: 10.1016/j.jde.2008.02.010.

[1]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[2]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133

[3]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641

[4]

Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations and Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403

[5]

Thierry Daudé, Damien Gobin, François Nicoleau. Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds. Inverse Problems and Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016

[6]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451

[7]

Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, Paweł Pilarczyk. Shadowing is generic---a continuous map case. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3591-3609. doi: 10.3934/dcds.2014.34.3591

[8]

Maciej J. Capiński. Covering relations and the existence of topologically normally hyperbolic invariant sets. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 705-725. doi: 10.3934/dcds.2009.23.705

[9]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[10]

Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787

[11]

Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795

[12]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307

[13]

Xiao Wen, Lan Wen. No-shadowing for singular hyperbolic sets with a singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6043-6059. doi: 10.3934/dcds.2020258

[14]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107

[15]

Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems and Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537

[16]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455

[17]

Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 687-698. doi: 10.3934/dcdsb.2005.5.687

[18]

Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6047-6072. doi: 10.3934/dcds.2018261

[19]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[20]

Juan Huang. Scattering and strong instability of the standing waves for dipolar quantum gases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4493-4513. doi: 10.3934/dcdsb.2020297

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (308)
  • HTML views (324)
  • Cited by (1)

Other articles
by authors

[Back to Top]