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Moser, J.

doi:10.1070/rd2001v006n03abeh000181

Cited by 9 publications

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“…Many dynamical features of exact symplectomorphisms of T * S 1 (they are also called globally area preserving mappings) are exhibited by arbitrary diffeomorphisms with the intersection property. In particular, for such diffeomorphisms, one can develop the KAM (Kolmogorov-Arnold-Moser) theory, see, e.g., the works [25,26,28,32]; for some recent results and generalizations see, e.g., the papers [18,19,23,35].…”

Section: The Intersection Property In Dimension Twomentioning

confidence: 99%

Three Examples in the Dynamical Systems Theory

Sevryuk¹

2022

SIGMA

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We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms R, S of a closed twodimensional annulus that possess the intersection property but their composition RS does not (R being just the rotation by π/2). The second example is that of a non-Lagrangian n-torus L 0 in the cotangent bundle T * T n of T n (n ≥ 2) such that L 0 intersects neither its images under almost all the rotations of T * T n nor the zero section of T * T n . The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form ẋ = f (x, y), ẏ = µg(x, y) in the closed upper half-plane {y ≥ 0} such that for each family, the corresponding phase portraits for 0 < µ < 1 and for µ > 1 are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.

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Section: The Intersection Property In Dimension Twomentioning

confidence: 99%

Three Examples in the Dynamical Systems Theory

Sevryuk¹

2022

SIGMA

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“…在解析函数范畴内, 函数导数的上界由函数值和解析区域的大小决定, 这使得坐标变换迭代收敛性的分析比较容易. 在二维保面积情形下, 运用光滑化技巧, Moser [39,40] 将光滑性降低至 333 阶可微. 在完成 Kolmogorov 定理证明的同时, Arnold [19] 也把相关理论应用到天体力学的一类 n 体问题的研究.…”

Section: Kolmogorov 定理unclassified

Dynamical diversity of nearly integrable Hamiltonian systems

Chong-Qing¹

2020

Sci. Sin.-Math.

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The dynamics of nearly integrable Hamiltonian is called by Poincaré the fundamental problem of dynamical systems. Since the fifties of the last century, the great achievements have been made in the study of the problem. Among them, the Kolmogorov's theorem on invariant tori and the discovery of Arnold diffusion are the landmarks. They have greatly deepened our understanding on the dynamical diversity of nearly integrable Hamiltonian systems. We shall briefly review some of the developments.

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Normal linear stability of quasi-periodic tori

Broer

Hoo

Naudot

2007

Journal of Differential Equations

113

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We consider families of dynamical systems having invariant tori that carry quasi-periodic motions. Our interest is the persistence of such tori under small, nearly-integrable perturbations. This persistence problem is studied in the dissipative, the Hamiltonian and the reversible setting, as part of a more general KAM theory for classes of structure preserving dynamical systems. This concerns the parametrized KAM theory as initiated by Moser [J.K. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (2) (1966)145-172; J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136-176] and further developed in [G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988; H.W. Broer, G.B. Huitema, F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (421) (1990) 1-82; H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1) (1995) 191-212]. The corresponding nondegeneracy condition involves certain (trans-)versality conditions on the normal linear, leading, part at the invariant tori. We show that as a consequence, a Cantor family of Diophantine tori with positive Hausdorff measure is persistent under nearly-integrable perturbations. This result extends the above references since presently the case of multiple Floquet exponents is included. Our leading example is the normal 1 : −1 resonance, which occurs a lot in applications, both Hamiltonian and reversible. As an illustration of this we briefly describe the Lagrange top coupled to an oscillator.

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Cited by 9 publications

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Three Examples in the Dynamical Systems Theory

Three Examples in the Dynamical Systems Theory

Dynamical diversity of nearly integrable Hamiltonian systems

Normal linear stability of quasi-periodic tori

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