On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

Wang, Xiaocai; Xu, Ji‐Qing

doi:10.1016/j.na.2007.08.016

Cited by 19 publications

(14 citation statements)

References 4 publications

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“…The idea is that if the non-degeneracy condition never occurs, then the non-resonant conditions are always the initial ones; if the non-degeneracy condition appears at some step, it can be kept on in the later steps. These particular phenomena [26,30] are the nature of 2-dimensional system, which don't hold in the high-dimensional case.…”

Section: Dongfeng Zhang Junxiang Xu and Xindong Xumentioning

confidence: 99%

“…In particular, there have been some interesting aspects for 2-dimensional quasiperiodic systems. Without imposing any non-degeneracy condition, the reducibility of two dimensional quasi-periodic system was obtained in [26,30]. The idea is that if the non-degeneracy condition never occurs, then the non-resonant conditions are always the initial ones; if the non-degeneracy condition appears at some step, it can be kept on in the later steps.…”

Section: Dongfeng Zhang Junxiang Xu and Xindong Xumentioning

confidence: 99%

“…Moreover, since K < Q n+2 , it follows that R Qn+2 B + (θ) = 0. By (37) and (38), we get the estimates (26) and (27).…”

Section: Lemma 22 (Step Lemma)mentioning

confidence: 99%

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Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Zhang¹,

Xu²,

Xu³

2018

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we consider the systemẋ = (A() + m P (t;))x, x ∈ R 3 , where is a small parameter, A, P are all 3 × 3 skew symmetric matrices, A is a constant matrix with eigenvalues ±iλ() and 0, whereλ() = λ + am 0 m 0 + O(m 0 +1)(m 0 < m), am 0 = 0, P is a quasi-periodic matrix with basic frequencies ω = (1, α) with α being irrational. First, it is proved that for most of sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that 0 ≤ β(α) < r, where β(α) measures how Liouvillean α is, r is the initial analytic radius, it is proved that for most of sufficiently small parameters, this system can be reduced to constant system by means of a quasi-periodic change of variables.

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Section: Dongfeng Zhang Junxiang Xu and Xindong Xumentioning

confidence: 99%

Section: Dongfeng Zhang Junxiang Xu and Xindong Xumentioning

confidence: 99%

See 1 more Smart Citation

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Zhang¹,

Xu²,

Xu³

2018

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

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“…Suppose that has different nonzero eigenvalues; they proved that, under some nonresonant conditions and nondegeneracy conditions, there exists a nonempty Cantor set ⊂ (0, 0 ), such that, for all ∈ , system (7) is reducible. Later, in [24], Wang and Xu considered the nonlinear quasiperiodic systeṁ=…”

Section: Introductionmentioning

confidence: 99%

On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

Xue

Zhao

2018

Journal of Function Spaces

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In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system x˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.

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“…where A has n different nonzero eigenvalues λ i and h = O(x 2 ). They proved that under some non-resonance conditions and some non-degeneracy conditions, there exists a non-empty Cantor subset E ⊂ (0, ε 0 ), such that the system (1.6) is reducible for ε ∈ E. Later, Wang and Xu [11] considered the analytic nonlinear quasi-periodic systemẋ = Ax…”

mentioning

confidence: 99%

On the reducibility of a class of almost periodic Hamiltonian systems

Li¹,

Xu²

2021

DCDS-B

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In this paper we consider the following linear almost periodic hamiltonian systemẋ = (A + εQ(t, ε))x, x ∈ R 2 , where A is a constant matrix with different eigenvalues, and Q(t, ε) is analytic almost periodic with respect to t and analytic with respect to ε. Without any non-degeneracy condition, we prove that the linear hamiltonian system is reducible for most of sufficiently small parameter ε by an almost periodic symplectic mapping. 2020 Mathematics Subject Classification. 37J40.

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On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

Cited by 19 publications

References 4 publications

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

On the reducibility of a class of almost periodic Hamiltonian systems

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