Naslov
Entropy of Lagrangian systems and invariant sets of extended gradient systems

Autori
Slijepčević, Siniša

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Dynamics of evolution equations / Raugel, Genevieve - , 2016, 15-15

Skup
Dynamics of Evolution Equations in memory of Jack Hale

Mjesto i datum
Marseille, Francuska, 21.03.2016. - 25.03.2016

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Entropy ; Lagrangian systems ; Extended gradient systems

Sažetak
We consider in parallel dynamical properties of finite dimensional Lagrangian systems, and properties of formally gradient dynamics of its action functional. The considered example is a family of Arnold’s-like Lagrangians in two and a half degrees of freedom, and the associate extended (or formally-) gradient system - a pair of coupled reaction- diffusion equations considered on an unbounded domain. (The techniques, though, seem to be applicable in much more general cases). For Lagrangian systems, we consider the problem of constructing “diffusion” orbits of infinite length, and positive entropy invariant measures. Our aim is towards developing a quantitative version of the variational approach to Arnold’s diffusion by Bernard, Cheng, Kaloshin, Yan and Zhang. We construct diffusion orbits and positive entropy measures whenever there is no topological obstruction to dif- fusion, and estimate the diffusion time and the topological entropy in terms of a certain bound
1 on the variation of the action, obtained by an application of the infinite- dimensional Morse-Sard theorem. The method seems optimal: for small values of the Arnold’s parameter  , we get topological entropy to be O (  ) , which corresponds to opti- mally “fast” diffusion orbits which change momentum by O (1) for the time O (  j log  j ) . The construction method is new, and relies on constructing lots of uniformly-local invariant sets for the associated evolutionary equations on an unbounded domain. We apply an abstract theory partly jointly developed with Thierry Gallay, and construct invariant sets by a precise study of the interplay of the “energy” dissipation and flux, energy in this case being the Lagrangian action.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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