{"abstract":[{"text":"The purpose of this paper is to construct examples of diffusion for E-Hamiltonian perturbations\r\nof completely integrable Hamiltonian systems in 2d-dimensional phase space, with d large.\r\nIn the first part of the paper, simple and explicit examples are constructed illustrating absence\r\nof ‘long-time’ stability for size E Hamiltonian perturbations of quasi-convex integrable systems\r\nalready when the dimension 2d of phase space becomes as large as log 1/E . We first produce\r\nthe example in Gevrey class and then a real analytic one, with some additional work.\r\nIn the second part, we consider again E-Hamiltonian perturbations of completely integrable\r\nHamiltonian system in 2d-dimensional space with E-small but not too small, |E| > exp(−d), with\r\nd the number of degrees of freedom assumed large. It is shown that for a class of analytic\r\ntime-periodic perturbations, there exist linearly diffusing trajectories. The underlying idea for\r\nboth examples is similar and consists in coupling a fixed degree of freedom with a large\r\nnumber of them. The procedure and analytical details are however significantly different. As\r\nmentioned, the construction in Part I is totally elementary while Part II is more involved, relying\r\nin particular on the theory of normally hyperbolic invariant manifolds, methods of generating\r\nfunctions, Aubry–Mather theory, and Mather’s variational methods.","lang":"eng"}],"status":"public","date_created":"2020-09-18T10:49:06Z","publication":"Journal of Functional Analysis","intvolume":"       229","date_updated":"2021-01-12T08:19:49Z","article_processing_charge":"No","quality_controlled":"1","date_published":"2005-12-01T00:00:00Z","year":"2005","day":"01","author":[{"last_name":"Bourgain","full_name":"Bourgain, Jean","first_name":"Jean"},{"id":"FE553552-CDE8-11E9-B324-C0EBE5697425","last_name":"Kaloshin","first_name":"Vadim","orcid":"0000-0002-6051-2628","full_name":"Kaloshin, Vadim"}],"extern":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"1-61","publisher":"Elsevier","publication_status":"published","volume":229,"issue":"1","title":"On diffusion in high-dimensional Hamiltonian systems","month":"12","_id":"8516","doi":"10.1016/j.jfa.2004.09.006","publication_identifier":{"issn":["0022-1236"]},"type":"journal_article","citation":{"short":"J. Bourgain, V. Kaloshin, Journal of Functional Analysis 229 (2005) 1–61.","apa":"Bourgain, J., &#38; Kaloshin, V. (2005). On diffusion in high-dimensional Hamiltonian systems. <i>Journal of Functional Analysis</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jfa.2004.09.006\">https://doi.org/10.1016/j.jfa.2004.09.006</a>","ama":"Bourgain J, Kaloshin V. On diffusion in high-dimensional Hamiltonian systems. <i>Journal of Functional Analysis</i>. 2005;229(1):1-61. doi:<a href=\"https://doi.org/10.1016/j.jfa.2004.09.006\">10.1016/j.jfa.2004.09.006</a>","ista":"Bourgain J, Kaloshin V. 2005. On diffusion in high-dimensional Hamiltonian systems. Journal of Functional Analysis. 229(1), 1–61.","chicago":"Bourgain, Jean, and Vadim Kaloshin. “On Diffusion in High-Dimensional Hamiltonian Systems.” <i>Journal of Functional Analysis</i>. Elsevier, 2005. <a href=\"https://doi.org/10.1016/j.jfa.2004.09.006\">https://doi.org/10.1016/j.jfa.2004.09.006</a>.","ieee":"J. Bourgain and V. Kaloshin, “On diffusion in high-dimensional Hamiltonian systems,” <i>Journal of Functional Analysis</i>, vol. 229, no. 1. Elsevier, pp. 1–61, 2005.","mla":"Bourgain, Jean, and Vadim Kaloshin. “On Diffusion in High-Dimensional Hamiltonian Systems.” <i>Journal of Functional Analysis</i>, vol. 229, no. 1, Elsevier, 2005, pp. 1–61, doi:<a href=\"https://doi.org/10.1016/j.jfa.2004.09.006\">10.1016/j.jfa.2004.09.006</a>."},"article_type":"original","keyword":["Analysis"],"oa_version":"None"}