Science China Mathematics

, Volume 62, Issue 11, pp 2103–2130 | Cite as

Variational approach to Arnold diffusion

  • Chong-Qing ChengEmail author
  • Jinxin Xue


Arnold diffusion was conjectured by Arnol’d (1964) as a typical phenomena of topological instability in classical mechanics. In this paper, we give a panorama of the researches on Arnold diffusion using the variational approaches.


Arnold diffusion Aubry set normal hyperbolicity normal form variational methods genericity 


37J40 37J50 49L25 



The first author was supported by National Natural Science Foundation of China (Grant Nos. 11790272 and No.11631006). The second author was supported by National Natural Science Foundation of China (Grant No. 11790273) and Beijing Natural Science Foundation (Grant No. Z180003).


  1. 1.
    Arnol’d V I. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math Surveys, 1963, 18: 85–192MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnol’d V I. Instability of dynamical systems with several degrees of freedom (Instability of motions of dynamic system with five-dimensional phase space). Soviet Math, 1964, 5: 581–585zbMATHGoogle Scholar
  3. 3.
    Arnol’d V I. Mathematical Methods of Classical Mechanics. New York: Springer, 1989CrossRefGoogle Scholar
  4. 4.
    Arnol’d V I. Mathematical problems in classical physics. In: Trends and Perspectives in Applied Mathematics. Applied Mathematical Sciences, vol. 100. New York: Springer, 1994, 1–20Google Scholar
  5. 5.
    Arnol’d V I. The stability problem and ergodic properties for classical dynamical systems. In: Proceedings of International Congress of Mathematicians. Berlin-Heidelberg: Springer, 2014, 107–113Google Scholar
  6. 6.
    Bangert V. Mather sets for twist maps and geodesics on tori. In: Dynamics Reported. Dynamics Reported, vol. 1. Wiesbaden: Vieweg+Teubner Verlag, 1988, 1–56Google Scholar
  7. 7.
    Bernard P. Symplectic aspects of Mather theory. Duke Math J, 2007, 136: 401–420MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bernard P. The dynamics of pseudographs in convex Hamiltonian systems. J Amer Math Soc, 2008, 21: 615–669MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bernard P, Contreras G. A generic property of families of Lagrangian systems. Ann of Math (2), 2008, 167: 1099–1108MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bernard P, Kaloshin V, Zhang K. Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders. Acta Math, 2017, 217: 1–79CrossRefGoogle Scholar
  11. 11.
    Cheng C-Q. Arnold diffusion in nearly integrable Hamiltonian systems. ArXiv:1207.4016, 2012. Permanent preprint. Decomposed and expanded into [12, 13, 20]Google Scholar
  12. 12.
    Cheng C-Q. Uniform hyperbolicity of invariant cylinder. J Differential Geom, 2017, 106: 1–43MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cheng C-Q. Dynamics around the double resonance. Cambridge J Math, 2017, 5: 153–228MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cheng C-Q. The genericity of Arnold diffusion in nearly integerable Hamiltonian systems. Asian J Math, 2019, in pressGoogle Scholar
  15. 15.
    Cheng C-Q, Xue J. Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom. ArX-iv:1503.04153v2, 2015Google Scholar
  16. 16.
    Cheng C-Q, Xue J. Order property and Modulus of Continuity of Weak KAM Solutions. Calc Var Partial Differential Equations, 2018, 57: 65MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cheng C-Q, Yan J. Existence of diffusion orbits in a priori unstable Hamiltonian systems. J Differential Geom, 2004, 67: 457–517MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cheng C-Q, Yan J. Arnold diffusion in Hamiltonian Systems: a priori Unstable Case. J Differential Geom, 2009, 82: 229–277MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cheng C-Q, Zhou M. Non-degeneracy of extremal points in multi-dimensional space. Sci China Math, 2015, 58: 2255–2260MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cheng C-Q, Zhou M. Hyperbolicity of minimal periodic orbits. Math Res Lett, 2016, 23: 685–705MathSciNetCrossRefGoogle Scholar
  21. 21.
    Delshams A, de la Llave R, Seara T M. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model. Mem Amer Math Soc, 2006, 179: 844MathSciNetzbMATHGoogle Scholar
  22. 22.
    Delshams A, de la Llave R, Seara T M. Geometric properties of the scattering map of a normally hyperbolic invariant manifold. Adv Math, 2008, 217: 1096–1153MathSciNetCrossRefGoogle Scholar
  23. 23.
    Delshams A, de la Llave R, Seara T M. Instability of high dimensional hamiltonian systems: Multiple resonances do not impede diffusion. Adv Math, 2016, 294: 689–755MathSciNetCrossRefGoogle Scholar
  24. 24.
    Delshams A, Huguet G. Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems. Nonlinearity, 2009, 22: 1997–2077MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fathi A. Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Adavnced Mathematics. Cambridge: Cambridge University Press, 2009Google Scholar
  26. 26.
    Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J, 1971, 21: 193–226MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gelfreich V, Turaev D. Arnold Diffusion in a priory chaotic Hamiltonian systems. ArXiv: 1406.2945, 2014Google Scholar
  28. 28.
    Gidea M, Marco J-P. Diffusion along chains of normally hyperbolic cylinders. ArXiv: 1708.08314, 2017Google Scholar
  29. 29.
    Hirsch M, Pugh C, Shub M. Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. New York: Springer-Verlag, 1977CrossRefGoogle Scholar
  30. 30.
    Kaloshin V, Zhang K. A strong form of Arnold diffusion for two and a half degrees of freedom. ArXiv:1212.1150v3, 2018Google Scholar
  31. 31.
    Kaloshin V, Zhang K. Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion. Bull Soc Math France, 2019, in pressGoogle Scholar
  32. 32.
    Li X. On c-equivalence. Sci China Ser A, 2009, 52: 2389–2396MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li X, Cheng C-Q. Connecting orbits of autonomous Lagrangian systems. Nonlinearity, 2009, 23: 119–141MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mañé R. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 1996, 9: 273–310MathSciNetCrossRefGoogle Scholar
  35. 35.
    Marco J-P. Chains of compact cylinders for cusp-generic nearly integrable convex systems on \(\mathbb{A}^{3}\). ArXiv:1602.02399, 2016Google Scholar
  36. 36.
    Mather J. Action minimizing invariant measures for positive definite Lagrangian systems. Math Z, 1991, 207: 169–207MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mather J. Variational construction of connecting orbits. Ann Inst Fourier (Grenoble), 1993, 43: 1349–1386MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mather J. Arnold diffusion, I: Announcement of results. J Math Sci, 2004, 124: 5275–5289MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pöschel J. A lecture on the classical KAM theorem. Proc Sympos Pure Math, 2001, 69: 707–732MathSciNetCrossRefGoogle Scholar
  40. 40.
    Treschev D V. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity, 2004, 17: 1803–1841MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniverisityNanjingChina
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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