Motivated by problems arising in nonlinear PDE's with a Hamiltonian structure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type $\ddot x=\dpr_x V$, [$x\in\TN$, $\dpr_x\=(\dpr_{x_1},..., \dpr_{x_N})$]. Extending methods from quantitative perturbation theory (Kolmogorov--Arnold--Moser theory, Nash--Moser implicit function theorem, \etc) we construct uncountably many almost--periodic solutions for the infinite dimensional system $\ddot x_i=f_i(x)$, $i\in \Zd$, $x\in \T^\Zd$ (endowed with the compact topology); the Hamiltonian structure is reflected by $f$ being a ``generalized gradient". Such result is derived under (suitable) analyticity assumptions on $f_i$ but without requiring any ``smallness conditions".

Chierchia, L., Perfetti, P. (1995). Second Order Hamiltonian Equations on $\T^\io$cand Almost--Periodic Solutions. JOURNAL OF DIFFERENTIAL EQUATIONS, 116, 172-201.

Second Order Hamiltonian Equations on $\T^\io$cand Almost--Periodic Solutions

PERFETTI, PAOLO
1995-02-01

Abstract

Motivated by problems arising in nonlinear PDE's with a Hamiltonian structure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type $\ddot x=\dpr_x V$, [$x\in\TN$, $\dpr_x\=(\dpr_{x_1},..., \dpr_{x_N})$]. Extending methods from quantitative perturbation theory (Kolmogorov--Arnold--Moser theory, Nash--Moser implicit function theorem, \etc) we construct uncountably many almost--periodic solutions for the infinite dimensional system $\ddot x_i=f_i(x)$, $i\in \Zd$, $x\in \T^\Zd$ (endowed with the compact topology); the Hamiltonian structure is reflected by $f$ being a ``generalized gradient". Such result is derived under (suitable) analyticity assumptions on $f_i$ but without requiring any ``smallness conditions".
feb-1995
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
KAM-Theorem
http://www.sciencedirect.com/science?_ob=ArticleListURL&_method=list&_ArticleListID=1883170742&_sort=r&_st=13&view=c&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=0d604571d6832684babe0b50ff539861&searchtype=a
http://www.sciencedirect.com/science/article/pii/S0022039685710339
Chierchia, L., Perfetti, P. (1995). Second Order Hamiltonian Equations on $\T^\io$cand Almost--Periodic Solutions. JOURNAL OF DIFFERENTIAL EQUATIONS, 116, 172-201.
Chierchia, L; Perfetti, P
Articolo su rivista
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/45639
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 12
  • ???jsp.display-item.citation.isi??? ND
social impact