scholarly journals A $C^\infty$ Nekhoroshev theorem

2021 ◽  
Vol 3 (2) ◽  
pp. 1-17
Author(s):  
Dario Bambusi ◽  
◽  
Beatrice Langella ◽  
◽  
◽  
...  
Keyword(s):  
Nekhoroshev Theorem

On the Diffusion Along Resonant Lines in Continuous and Discrete Dynamical Systems

2003 ◽  
Vol 17 (22n24) ◽  
pp. 3964-3976
Author(s):  
Claude Froeschlé ◽  
Elena Lega
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Coupling Parameter ◽  
A Priori ◽  
Integrable Hamiltonian System ◽  
Discrete Dynamical Systems ◽  
Nekhoroshev Theorem ◽  
Symplectic Map ◽  
Quasi Integrable Hamiltonian System

We detect and measure diffusion along resonances in a discrete symplectic map for different values of the coupling parameter. Qualitatively and quantitatively the results are very similar to those obtained for a quasi-integrable Hamiltonian system, i.e. in agreement with Nekhoroshev predictions, although the discrete mapping does not fulfill completely, a priori, the conditions of the Nekhoroshev theorem.

scholarly journals Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

2014 ◽  
Vol 21 (1) ◽  
pp. 165-185 ◽  
Author(s):  
S. Wiggins ◽  
A. M. Mancho
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Fluid Transport ◽  
Invariant Tori ◽  
Point Of View ◽  
Time Dependent ◽  
Time Interval ◽  
Two Dimensional ◽  
Kam Theorem ◽  
Nekhoroshev Theorem

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

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