Book contents
- Frontmatter
- Contents
- Denjoy subsystems and horseshoes
- Impact Hamiltonian systems and polygonal billiards
- Some remarks on the classical KAM theorem, following Pöschel
- Some recent developments in Arnold diffusion
- Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold
- Holonomy and vortex structures in quantum hydrodynamics
- Surfaces of locally minimal flux
- A symplectic approach to Arnold diffusion problems
- Hamiltonian ODE, homogenization, and symplectic topology
- References
A symplectic approach to Arnold diffusion problems
Published online by Cambridge University Press: 10 May 2024
Book contents
- Frontmatter
- Contents
- Denjoy subsystems and horseshoes
- Impact Hamiltonian systems and polygonal billiards
- Some remarks on the classical KAM theorem, following Pöschel
- Some recent developments in Arnold diffusion
- Viscosity solutions of the Hamilton–Jacobi equation on a noncompact manifold
- Holonomy and vortex structures in quantum hydrodynamics
- Surfaces of locally minimal flux
- A symplectic approach to Arnold diffusion problems
- Hamiltonian ODE, homogenization, and symplectic topology
- References
Summary
The purpose of this text is to present a symplectic approach to Arnold diffusion problems, that is, the existence of orbits of perturbed integrable systems along which the action variables experience a drift whose length is independent of the size of the perturbation. We choose to focus on the construction of orbits drifting along chains of cylinders, taking for granted the existence of the chains. We however give a rather complete description of these chains, together with some elements on their symplectic features and some main ideas to prove their existence. We adopt the setting introduced by John Mather to prove the Arnold conjecture for perturbations of Tonelli Hamiltonians, which we see as the appropriate one to set out the various (and numerous) problems of the construction, and give some ideas to show how the symplectic approach may enable one to enlarge its scope.
Keywords
- Type
- Chapter
- Information
- Hamiltonian SystemsDynamics, Analysis, Applications, pp. 229 - 296Publisher: Cambridge University PressPrint publication year: 2024
References
Arnaud, M.-C. and Xue, J., “A C1 Arnold–Liouville theorem”, pp. 1–31 Astérisque 416, Société Mathématique de France, Paris, 2020.Google Scholar
Arnold, V., “First steps in symplectic topology”, Russian Mathematical Surveys 41:6 (1986), 1.CrossRefGoogle Scholar
Barbieri, S., Marco, J.-P., and Massetti, J. E., “Flexibility and analytic smoothing in averaging theory”, preprint, 2021. arXiv 2105.08365Google Scholar
Berger, P. and Bounemoura, A., “A geometrical proof of the persistence of normally hyperbolic submanifolds”, Dyn. Syst. 28:4 (2013), 567–581.CrossRefGoogle Scholar
Bernard, P., “The dynamics of pseudographs in convex Hamiltonian systems”, J. Amer. Math. Soc. 21:3 (2008), 615–669.CrossRefGoogle Scholar
Bernard, P., “Large normally hyperbolic cylinders in a priori stable Hamiltonian systems”, Ann. Henri Poincaré 11:5 (2010), 929–942.CrossRefGoogle Scholar
Bernard, P., Kaloshin, V., and Zhang, K., “Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders”, Acta Math. 217:1 (2016), 1–79.CrossRefGoogle Scholar
Bolotin, S. and Treschev, D., “Unbounded growth of energy in nonautonomous Hamiltonian systems”, Nonlinearity 12:2 (1999), 365–388.CrossRefGoogle Scholar
Chaperon, M., “The Lipschitzian core of some invariant manifold theorems”, Ergodic Theory Dynam. Systems 28:5 (2008), 1419–1441.CrossRefGoogle Scholar
Cheng, C.-Q., “The genericity of Arnold diffusion in nearly integrable Hamiltonian systems”, Asian J. Math. 23:3 (2019), 401–438.CrossRefGoogle Scholar
Cheng, C.-Q. and Yan, J., “Arnold diffusion in Hamiltonian systems: a priori unstable case”, J. Differential Geom. 82:2 (2009), 229–277.CrossRefGoogle Scholar
Delshams, A. and Huguet, G., “Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems”, Nonlinearity 22:8 (2009), 1997–2077.CrossRefGoogle Scholar
Delshams, A. and Schaefer, R. G., “Arnold diffusion for a complete family of perturbations”, Regul. Chaotic Dyn. 22:1 (2017), 78–108.CrossRefGoogle Scholar
Delshams, A., de la Llave, R., and Seara, T. M., “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. 209:2 (2000), 353–392.CrossRefGoogle Scholar
Delshams, A., de la Llave, R., and Seara, T. M., “A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model”, 179:844 (2006), viii+141 in.CrossRefGoogle Scholar
Delshams, A., de la Llave, R., and Seara, T. M., “Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows”, Adv. Math. 202:1 (2006), 64–188.CrossRefGoogle Scholar
Delshams, A., de la Llave, R., and Seara, T. M., “Geometric properties of the scattering map of a normally hyperbolic invariant manifold”, Adv. Math. 217:3 (2008), 1096–1153.CrossRefGoogle Scholar
Fathi, A., “Weak KAM theorem in Lagrangian dynamics”, preliminary version, number 10, 2008, tinyurl.com/FathiKAM.Google Scholar
Gelfreich, V. and Turaev, D., “Arnold diffusion in a priori chaotic symplectic maps”, Comm. Math. Phys. 353:2 (2017), 507–547.CrossRefGoogle Scholar
Gidea, M. and de la Llave, R., “Topological methods in the instability problem of Hamiltonian systems”, Discrete Contin. Dyn. Syst. 14:2 (2006), 295–328.CrossRefGoogle Scholar
Gidea, M. and Marco, J.-P., “Diffusion along chains of normally hyperbolic cylinders”, preprint, 2017. arXiv 1708.08314Google Scholar
Gidea, M. and Robinson, C., “Diffusion along transition chains of invariant tori and Aubry– Mather sets”, Ergodic Theory Dynam. Systems 33:5 (2013), 1401–1449.CrossRefGoogle Scholar
Gidea, M., de la Llave, R., and T. M-Seara, “A general mechanism of diffusion in Hamiltonian systems: qualitative results”, Comm. Pure Appl. Math. 73:1 (2020), 150–209.CrossRefGoogle Scholar
Gidea, M., de la Llave, R., and Seara, T. M., “A general mechanism of instability in Hamiltonian systems: skipping along a normally hyperbolic invariant manifold”, Discrete Contin. Dyn. Syst. 40:12 (2020), 6795–6813.CrossRefGoogle Scholar
Givental, A. B., Khesin, B. A., Marsden, J. E., Varchenko, A. N., Vassiliev, V. A., Viro, O. Y., and Zakalyukin, V. M. (editors), Instability of dynamical systems with several degrees of freedom, Springer, Berlin, Heidelberg, 2009.Google Scholar
Gromov, M., “Pseudo holomorphic curves in symplectic manifolds”, Invent. Math. 82:2 (1985), 307–347.CrossRefGoogle Scholar
Haller, G., “Diffusion at intersecting resonances in Hamiltonian systems”, Phys. Lett. A 200:1 (1995), 34–42.CrossRefGoogle Scholar
Haller, G., “Universal homoclinic bifurcations and chaos near double resonances”, J. Statist. Phys. 86:5-6 (1997), 1011–1051.CrossRefGoogle Scholar
Herman, M.-R., Sur les courbes invariantes par les difféomorphismes de l’anneau, Vol. 1, Astérisque 103, Société Mathématique de France, Paris, 1983. With an appendix by Albert Fathi.Google Scholar
Herman, M.-R., Sur les courbes invariantes par les difféomorphismes de l’anneau, Vol. 2, Astérisque 144, Société Mathématique de France, Paris, 1986.Google Scholar
Hirsch, M. W., Pugh, C. C., and Shub, M., Invariant manifolds, Lecture Notes in Mathematics 583, Springer, 1977.Google Scholar
Kaloshin, V. and Saprykina, M., “An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension”, Comm. Math. Phys. 315:3 (2012), 643–697.CrossRefGoogle Scholar
Kaloshin, V. and Zhang, K., Arnold diffusion for smooth systems of two and a half degrees of freedom, Annals of Mathematics Studies 208, Princeton University Press, 2020.Google Scholar
Katok, A. and Hasselblatt, B., Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995.Google Scholar
Koropecki, A. and Nassiri, M., “Transitivity of generic semigroups of area-preserving surface diffeomorphisms”, Math. Z. 266:3 (2010), 707–718.CrossRefGoogle Scholar
Koropecki, A. and Nassiri, M., “Erratum to: Transitivity of generic semigroups of area-preserving surface diffeomorphisms [MR2719428]”, Math. Z. 268:1-2 (2011), 601–604.CrossRefGoogle Scholar
Lalonde, F. and Sikorav, J.-C., “Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents”, Comment. Math. Helv. 66:1 (1991), 18–33.CrossRefGoogle Scholar
Laudenbach, F. and Sikorav, J.-C., “Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent”, Invent. Math. 82:2 (1985), 349–357.CrossRefGoogle Scholar
Le Calvez, P., “Propriétés dynamiques des régions d’instabilité”, Ann. Sci. École Norm. Sup. (4) 20:3 (1987), 443–464.CrossRefGoogle Scholar
Le Calvez, P., “Propriétés dynamiques des difféomorphismes de l’anneau et du tore”, Asté-risque 204 (1991), 131.Google Scholar
Le Calvez, P., “Drift orbits for families of twist maps of the annulus”, Ergodic Theory Dynam. Systems 27:3 (2007), 869–879.CrossRefGoogle Scholar
Mandorino, V., “Generic transitivity for couples of Hamiltonians”, J. Dyn. Control Syst. 19:4 (2013), 593–610.CrossRefGoogle Scholar
Marco, J.-P., “Arnold diffusion for cusp-generic nearly integrable convex systems on 𝔸3”, preprint, 2016. arXiv 1602.02403Google Scholar
Marco, J.-P., “Chains of compact cylinders for cusp-generic nearly integrable convex systems on 𝔸3”, preprint, 2016. arXiv 1602.02399Google Scholar
Marco, J.-P., “Twist maps and Arnold diffusion for diffeomorphisms”, pp. 473–495 in Variational methods, edited by Bergounioux, M. et al., Radon Ser. Comput. Appl. Math. 18, De Gruyter, Berlin, 2017.Google Scholar
McDuff, D. and Salamon, D., Introduction to symplectic topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.Google Scholar
Mèzer, D. N., “Arnold diffusion, I: Announcement of results”, Sovrem. Mat. Fundam. Napravl. 2 (2003), 116–130.Google Scholar
Moeckel, R., “Generic drift on Cantor sets of annuli”, pp. 163–171 in Celestial mechanics (Evanston, IL, 1999), edited by Chenciner, A. et al., Contemp. Math. 292, Amer. Math. Soc., Providence, RI, 2002.Google Scholar
Nassiri, M. and Pujals, E. R., “Robust transitivity in Hamiltonian dynamics”, Ann. Sci. Éc. Norm. Supér. (4) 45:2 (2012), 191–239.CrossRefGoogle Scholar
Robinson, R. C., “Generic properties of conservative systems”, Amer. J. Math. 92 (1970), 562–603.CrossRefGoogle Scholar
Robinson, R. C., “Generic properties of conservative systems, II”, Amer. J. Math. 92 (1970), 897–906.CrossRefGoogle Scholar
Sabbagh, L., “An inclination lemma for normally hyperbolic manifolds with an application to diffusion”, Ergodic Theory Dynam. Systems 35:7 (2015), 2269–2291.CrossRefGoogle Scholar
Treschev, D., “Evolution of slow variables in a priori unstable Hamiltonian systems”, Nonlinearity 17:5 (2004), 1803–1841.CrossRefGoogle Scholar
Yoccoz, J.-C., “Travaux de Herman sur les tores invariants”, exposé 754, pp. 311–344 in Séminaire Bourbaki, 1991/92, Astérisque 206, Société Mathématique de France, Paris, 1992.Google Scholar