L. Chierchia et P. Perfetti, 2ND-ORDER HAMILTONIAN EQUATIONS ON T-INFINITY AND ALMOST-PERIODIC SOLUTIONS, Journal of differential equations, 116(1), 1995, pp. 172-201
Motivated by problems arising in nonlinear PDE's with a Hamiltonian st
ructure and in high dimensional dynamical systems, we study a suitable
generalization to infinite dimensions of second order Hamiltonian equ
ations of the type ($) over bar x=partial derivative(x)V, [x is an ele
ment of T-N, partial derivative(x)=(partial derivative(x1), ..., parti
al derivative(xN))]. Extending methods from quantitative perturbation
theory (Kolmogorov-Arnold-Moser theory, Nash-Moser implicit function t
heorem, etc.) we construct uncountably many almost-periodic solutions
for the infinite dimensional system ($) over bar x(i)=f(i)(x), i is an
element of Z(d), x is an element of T-Zd (endowed with the compact to
pology); the Hamiltonian structure is reflected by f being a ''general
ized gradient.'' Such a result is derived under (suitable) analyticity
assumptions on f(i) but without requiring any ''smallness conditions.
'' (C) 1995 Academic Press, Inc.