Non-degeneracy of extremal points in multi-dimensional space

Cheng, Chuan; Zhou, Min

doi:10.1007/s11425-015-5057-6

Cited by 6 publications

(3 citation statements)

References 5 publications

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“…Because the set of double resonant points is dense along the resonant path, it appears necessary to ask whether it holds simultaneously for all p ∈ Γ ′ that the minimal point of Z k ′ (p, x) is non-degenerate when it is treated as a function of x = k ′ , q . Fortunately, we have the following result [CZ2] Theorem 6.1. Assume M is a closed manifold with finite dimensions, F ζ ∈ C r (M, R) with r ≥ 4 for each ζ ∈ [ζ 0 , ζ 1 ] and F ζ is Lipschitz in the parameter ζ.…”

Section: Criterion For Strong and Weak Double Resonancementioning

confidence: 92%

Uniform hyperbolicity of invariant cylinder

Cheng¹

2017

J. Differential Geom.

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Section: Criterion For Strong and Weak Double Resonancementioning

confidence: 92%

Uniform hyperbolicity of invariant cylinder

Cheng¹

2017

J. Differential Geom.

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“…Because the set of double resonant points is dense along the resonant path, it appears necessary to ask whether it holds simultaneously for all p ∈ Γ ′ that the minimal point of Z k ′ (p, x) is non-degenerate when it is treated as a function of x = k ′ , q . Fortunately, we have the following result [CZ2] Theorem 6.1. Assume M is a closed manifold with finite dimensions, F So, once one has a generic single resonant term Z k ′ , the non-degeneracy λ is lower bounded from zero for all double resonant points.…”

Section: Criterion For Strong and Weak Double Resonancementioning

confidence: 92%

Uniform hyperbolicity of invariant cylinder

Cheng¹

2015

Preprint

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For a nearly integrable Hamiltonian systems H = h(p) + ǫP (p, q) with (p, q) ∈ R 3 × T 3 , large normally hyperbolic invariant cylinders exist along the whole resonant path, except for the √ ǫ 1+d -neighborhood of finitely many double resonant points. It allows one to construct diffusion orbits to cross double resonance.[Z2] Zhou M., Infinity of minimal homoclinic orbits, Nonlinearity 24 (2011) 931-939.

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“…Proof. The statement (without the "Moreover" part) can be obtained directly by applying the main theorem of [CZ2] which is a higher dimensional generalization of Proposition 4.7. Here we give an argument using only Proposition 4.7.…”

Section: Dynamics Around Triple Resonancesmentioning

confidence: 99%

Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom

Cheng,

Xue

2015

Preprint

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In this paper Arnold diffusion is proved to be a generic phenomenon in the smooth categroy for nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom:Under typical perturbation εP , the system admits "connecting" orbit that passes through any finitely many prescribed small balls in the same energy levelContents 1. Introduction 2. Proof of the main theorem 2.1. Conventions and notations 2.2. The choice of the frequency path 2.3. The abstract variational framework 2.4. Existence of the generalized transition chain 3. The frequency segment and the KAM normal form 3.1. Number theoretic properties of the frequency line 3.2. Resonant submanifolds and their neighborhoods 3.3. Homogenization 3.4. The KAM normal form 4. The reduction of order for single resonances 4.1. Normally hyperbolic invariant manifold for Hamiltonian system 4.2. Normally hyperbolic invariant cylinder (NHIC) around single resonances 5. Dynamics around strong double resonances 5.1. Distinguishing weak and strong double resonances 5.2. The shear transformation for strong double resonances 5.3. Hamiltonian systems of two degrees of freedom 5.4. The high energy regime 5.5. Cohomology equivalence around strong double resonances 5.6. The generalized transition chain mapped to the frequency space 5.7. Center straightening 6. Dynamics around triple resonances 6.1. Frequency refinement 6.2. The nondegeneracy condition

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Non-degeneracy of extremal points in multi-dimensional space

Cited by 6 publications

References 5 publications

Uniform hyperbolicity of invariant cylinder

Uniform hyperbolicity of invariant cylinder

Uniform hyperbolicity of invariant cylinder

Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom

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