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

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

“…In the first step, we proved that has 2 N different eigenvalues and and are smaller perturbations. Now the KAM method will be used to prove Theorem 2.1 and we will use a similar process as that in [ 5 ] and [ 8 ]. For simplification of notations, here denotes the different constants.…”

“…In 2015, Li et al [ 8 ] considered the following analytic quasi-periodic Hamiltonian system: where the constant matrix A has multiple eigenvalues, Q , g , and h are quasi-periodic with respect to t and ( ). They proved that by using the non-resonant conditions, non-degeneracy conditions, and a suitable hypothesis of analyticity, the Hamiltonian system ( 7 ) can be changed to another Hamiltonian system with an equilibrium by a q-p symplectic transformation.…”

“…To take over the difficulty from the infinite frequency which generates the small divisors problem, we need a stronger norm. Inspired by the works of [ 4 , 5 ], and [ 8 ], in this paper, we allow Q , g , and h to be the classes of almost-periodic matrices. Our study is about the reducibility of almost-periodic Hamiltonian systems to [ 4 ] and [ 8 ].…”

“…Inspired by the works of [ 4 , 5 ], and [ 8 ], in this paper, we allow Q , g , and h to be the classes of almost-periodic matrices. Our study is about the reducibility of almost-periodic Hamiltonian systems to [ 4 ] and [ 8 ]. So, the usual LP transformation for KAM iteration should not only be almost-periodic but also symplectic, which preserves the Hamiltonian structure.…”

On the reducibility of a class of almost periodic Hamiltonian systems

“…Without any non-degeneracy condition, they proved that one of two results holds true: (1) the system (1.7) is reducible toẏ = By + O(y) for ∀ε ∈ (0, ε 0 ); (2) there exists a non-empty Cantor subset E, such that the the system (1.7) is reducible toẏ = By + O(y 2 ) for ε ∈ E. Recently, under non-resonance conditions and nondegeneracy conditions, Li, Zhu and Chen [8] considered the reducibility of the 2ndimensional nonlinear quasi-periodic hamiltonian system with multiple eigenvalues. Almost periodic case is natural extension of quasi-periodic case, the difficulty is from the small divisors of infinitely many frequencies.…”