Reducible problem for a class of almost-periodic non-linear Hamiltonian systems

摘要

This paper studies the reducibility of almost-periodic Hamiltonian systems with small perturbation near the equilibrium which is described by the following Hamiltonian system: $$rac{dx}{dt} = J igl[{A} +arepsilon{Q}(t,arepsilon) igr]x+ arepsilon g(t,arepsilon)+h(x,t,arepsilon). $$ It is proved that, under some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity and for the sufficiently small ?μ, the system can be reduced to a constant coefficients system with an equilibrium by means of an almost-periodic symplectic transformation.KeywordsAlmost-periodic matrix??Reducibility??KAM iteration??Hamiltonian systems??Small divisors??1 IntroductionIn this paper we are studying the reducibility of the following almost-periodic Hamiltonian system: $$egin{aligned} rac{dx}{dt} = J igl[{A} + arepsilon{Q}(t,arepsilon) igr]x +arepsilon g(t,arepsilon)+h(x,t,arepsilon),quad xin mathbf{R}^{2N}, end{aligned}$$ (1) where J is an anti-symmetric symplectic matrix, A is a (2N imes 2N) symmetric constant matrix with possible multiple eigenvalues, and (Q(t)) is an analytic almost-periodic symmetric (2N imes2N) matrix with respect to t, (g(t,arepsilon)) and (h(x,t,arepsilon)) are almost-periodic 2N-dimensional vector-valued functions with respect to t, with basic frequencies (omega=(omega_{1},omega_{2},ldots)) and (h(x,t)=O(x^{2})) ((xightarrow0)), and $$J=left ( egin{matrix} 0 & I_{N} -I_{N} & 0 end{matrix} ight ) , $$ where (I_{N}) is a (Nimes N) identity matrix and ?μ is a sufficiently small parameter. First of all we will recall some previous results in the field of reducibility for analytic differential systems. Consider the differential equation $$egin{aligned} rac{dx}{dt} = A(t)x,quad {xinmathbf{R}^{m}}, end{aligned}$$ (2) where (A(t)) is an almost-periodic matrix. We call the transformation (x= P(t)y) almost-periodic Lyapunova??Perron (L-P) transformation, if (P(t)) is non-singular and P, ({P}^{-1}), and á1? are almost periodic. The transformed equation is $$egin{aligned} rac{dy}{dt} = {C}(t)y, end{aligned}$$ (3) where ({C}= {P}^{-1}(AP-dot{P})). If there exists an almost-periodic L-P transformation such that ({C}(t)) is a constant matrix, then we call equation (2) reducible.In recent years, many researchers have devoted themselves to the study of the reducibility of finite dimensional systems by means of the KAM methods. The well-known Floquet theorem states that every periodic differential equation (2) can be reduced to a constant coefficients differential equation (3) by means of a periodic change of variables with the same period as ({A}(t)). But, if ({A}(t)) is quasi-periodic (q-p), then there is an example in [1] which illustrates that (2) is irreducible. In 1981, Johnson and Sell [2] showed that if (A(t)) the quasi-periodic matrix satisfies a??full spectruma?? conditions, then (2) is reducible. In 1992, Jorba and Sim?3 [3] proved the reducibility result of linear quasi-periodic systems like (5) for the constant matrix A with distinct eigenvalues. In 1999, Xu [4] proved the reducibility result of linear quasi-periodic systems like (5) for the constant matrix A with multiple eigenvalues. In 1996, Jorba and Sim?3 [5] considered the quasi-periodic system $$ rac{dx}{dt} = igl[ {A}+arepsilon{Q}(t) igr] x+ arepsilon {g}(t)+ {h}(x,t),quad {xinmathbf{R}^{m},} $$ (4) where the constant matrix A has distinct eigenvalues. They proved that system (4) is reducible for (arepsilonin E) using the non-resonant conditions and non-degeneracy conditions, where E is the non-empty Cantor subset such that (Esubset(0,arepsilon_{0})). Instead of quasi-periodic reduction to a constant coefficient linear systems, in 1996, Xu and You [6] proved the reducibility of the linear almost-periodic differential equation $$ rac{dx}{dt} = igl[ {A}+arepsilon{Q}(t) igr] x,quad {x inmathbf{R}^{m},} $$ (5) where the constant matrix A has different eigenvalues and ({Q}(t)) is an (m imes m) analytic almost-periodic matrix with frequencies (omega= (omega_{1},omega_{2},ldots)). Under some small divisor conditions and for most sufficiently small ?μ, they proved that system (5) is reducible to the constant coefficient system by an affine almost-periodic transformation. In 2013, Qiu and Li [7] considered the following non-linear almost-periodic differential equation: $$ rac{dx}{dt} = igl[ {A}+arepsilon{a}(t) igr] {x}^{2n+1} + {h}(x,t, arepsilon)+ {f}(x,t,arepsilon),quad {xin mathbf{R},} $$ (6) where ({ngeq0}) is an integer, A is a positive number, ?μ is a small parameter, h is a higher order term, and f is a small perturbation term. They proved that under some suitable conditions and using the KAM method system (6) can be reduced to a suitable normal form with zero as an equilibrium point by an affine almost-periodic transformation, so it has a