On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter

and Applied Analysis 3 where A is an n × n constant matrix with n different eigenvalues λ1, λ2, . . . , λn and Q t is analytic quasi-periodic with respect to t with frequencies ω ω1, ω2, . . . , ωl . Here ε is a small perturbation parameter. Suppose that the following nonresonance conditions hold: ∣∣〈k,ω〉 √ −1 λi − λj ∣∣ ≥ α |k| , 1.8 for all k ∈ Z \ {0}, where α > 0 is a small constant and τ > l − 1. Assume that λj ε j 1, 2, . . . , n are eigenvalues of A ε Q . If the following non-degeneracy conditions hold: d dε ( λi ε − λj ε ∣∣∣∣ ε 0 / 0, ∀i / j, 1.9 then authors proved that for sufficiently small ε0 > 0, there exists a nonempty Cantor subset E ⊂ 0, ε0 , such that for ε ∈ E, the system 1.7 is reducible. Moreover, meas 0, ε0 \ E o ε0 . Some related problems were considered by Eliasson in 4, 5 . In the paper 4 , to study one-dimensional linear Schrödinger equation d2q dt2 Q ωt q Eq, 1.10 Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system: ṗ E −Q ωt q, q̇ p, 1.11 whereQ is an analytic quasi-periodic function and E is an energy parameter. The result in 4 implies that for almost every sufficiently large E, the quasi-periodic system 1.11 is reducible. Later, in 5 the author considered the almost reducibility of linear quasi-periodic systems. Recently, the similar problem was considered by Her and You 6 . Let C Λ, gl m,C be the set ofm×mmatricesA λ depending analytically on a parameter λ in a closed intervalΛ ⊂ R. In 6 , Her and You considered one-parameter families of quasi-periodic linear equations ẋ ( A λ g ω1t, . . . , ωlt, λ ) x, 1.12 where A ∈ C Λ, gl m,C , and g is analytic and sufficiently small. They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists an open and dense set A in C Λ, gl m,C , such that for each A ∈ A, the system 1.12 is reducible for almost all λ ∈ Λ. In 1996, Jorba and Simó extended the conclusion of the linear system to the nonlinear case. In 7 , Jorba and Simó considered the quasi-periodic system ẋ A εQ t x εg t h x, t , x ∈ R, 1.13 4 Abstract and Applied Analysis where A has n different nonzero eigenvalues λi. They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists a nonempty Cantor subset E ⊂ 0, ε0 , such that the system 1.13 is reducible for ε ∈ E. In 8 , the authors found that the non-degeneracy condition is not necessary for the two-dimensional quasi-periodic system. They considered the two-dimensional nonlinear quasi-periodic system: ẋ Ax f x, t, ε , x ∈ R2, 1.14 where A has a pair of pure imaginary eigenvalues ±−1ω0 with ω0 / 0 satisfying the nonresonance conditions |〈k,ω〉| ≥ α |k| , |〈k,ω〉 − 2ω0| ≥ α |k| 1.15 for all k ∈ Z \ {0}, where α > 0 is a small constant and τ > l − 1. Assume that f 0, t, ε O ε and ∂xf 0, t, ε O ε as ε → 0. They proved that either of the following two results holds: 1 for ∀ε ∈ 0, ε0 , the system 1.14 is reducible to ẏ By O y as y → 0; 2 there exists a nonempty Cantor subset E ⊂ 0, ε0 , such that for ε ∈ E the system 1.14 is reducible to ẏ By O y2 as y → 0. Note that the result 1 happens when the eigenvalue of the perturbed matrix of A in KAM steps has nonzero real part. But the authors were interested in the equilibrium of the transformed system and obtained a small quasi-periodic solution for the original system. Motivated by 8 , in this paper we consider the Hamiltonian system and we have a better result. 2. Main Results Theorem 2.1. Consider the following real two-dimensional Hamiltonian system