On the topology of nearly-integrable Hamiltonians at simple resonances

@article{Biasco2020OnTT,
  title={On the topology of nearly-integrable Hamiltonians at simple resonances},
  author={Luca Biasco and Luigi Chierchia},
  journal={Nonlinearity},
  year={2020}
}
We show that, in general, averaging at simple resonances a real--analytic, nearly--integrable Hamiltonian, one obtains a one--dimensional system with a cosine--like potential; ``in general'' means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; ``cosine--like'' means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. \\ Furthermore… Expand
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References

SHOWING 1-10 OF 46 REFERENCES
Lagrangian tori near resonances of near-integrable Hamiltonian systems
In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limitExpand
Prevalence of exponential stability among nearly integrable Hamiltonian systems
  • L. Niederman
  • Mathematics
  • Ergodic Theory and Dynamical Systems
  • 2007
In the 1970s, Nekhorochev proved that, for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long timeExpand
Rigorous estimates for the series expansions of Hamiltonian perturbation theory
In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory,Expand
Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems
We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrableExpand
Drift and diffusion in phase space
The problem of stability of action variables (i.e. of the adiabatic invariants) in perturbations of completely integrable (real analytic) hamiltonian systems with more than two degrees of freedom isExpand
Diffusion and stability in perturbed non-convex integrable systems
The Nekhoroshev theorem has become an important tool for explaining the long-term stability of many quasi-integrable systems of interest in physics. The action variables of systems that satisfy theExpand
Estimates in the kolmogorov theorem on conservation of conditionally periodic motions
Abstract A Hamiltonian system which differs from the integrable system by a small perturbation is considered. According to the Kolmogorov theorem /1–3/ the majority of invariant tori present in theExpand
Existence of Diffusion Orbits in a priori Unstable Hamiltonian Systems
Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and time-periodic Hamiltonian systems with two degrees of freedom. 1, Introduction and Results By theExpand
Effective Stability and KAM Theory
Abstract The two main stability results for nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem, concerning exponential lower bounds for the stability time (effective stability),Expand
A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems
In the present paper we give a proof of Nekhoroshev's theorem, which is concerned with an exponential estimate for the stability times in nearly integrable Hamiltonian systems. At variance with theExpand
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