@article{8689,
abstract = {This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the Whitney conjugacy of a non{degenerate, real{analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov's set are provided in the case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the d-torus and (B) a domain with C2 boundary times the d-torus. All constants are explicitly given.},
author = {Chierchia, Luigi and Koudjinan, Edmond},
issn = {1560-3547},
journal = {Regular and Chaotic Dynamics},
keywords = {Nearly{integrable Hamiltonian systems, perturbation theory, KAM Theory, Arnold's scheme, Kolmogorov's set, primary invariant tori, Lagrangian tori, measure estimates, small divisors, integrability on nowhere dense sets, Diophantine frequencies.},
number = {1},
pages = {61--88},
publisher = {Springer Nature},
title = {{V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates}},
doi = {10.1134/S1560354721010044},
volume = {26},
year = {2021},
}
@unpublished{9435,
abstract = {For any given positive integer l, we prove that every plane deformation of a circlewhich preserves the 1/2and 1/ (2l + 1) -rational caustics is trivial i.e. the deformationconsists only of similarities (rescalings and isometries).},
author = {Kaloshin, Vadim and Koudjinan, Edmond},
title = {{Non co-preservation of the 1/2 and 1/(2l+1)-rational caustics along deformations of circles}},
year = {2021},
}
@article{8691,
abstract = {Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.},
author = {Koudjinan, Edmond},
issn = {0022-0396},
journal = {Journal of Differential Equations},
keywords = {Analysis},
number = {6},
pages = {4720--4750},
publisher = {Elsevier},
title = {{A KAM theorem for finitely differentiable Hamiltonian systems}},
doi = {10.1016/j.jde.2020.03.044},
volume = {269},
year = {2020},
}
@article{8694,
abstract = {We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps fa(x)=a−x2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition 𝒫 of the parameter interval Ω=[1.4,2] into almost 4×106 subintervals, for each of which we compute to high precision the orbits of the critical points up to some time N and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide 𝒫 into a family 𝒫+ of intervals, which we call stochastic intervals, and a family 𝒫− of intervals, which we call regular intervals. We numerically prove that each interval ω∈𝒫+ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in ω has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in 𝒫+ are stochastic and most parameters belonging to the intervals in 𝒫− are regular, thus the names. We prove that the intervals in 𝒫+ occupy almost 90% of the total measure of Ω. The software and the data are freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parameterized families of dynamical systems.},
author = {Golmakani, Ali and Koudjinan, Edmond and Luzzatto, Stefano and Pilarczyk, Pawel},
journal = {Chaos},
number = {7},
publisher = {AIP},
title = {{Rigorous numerics for critical orbits in the quadratic family}},
doi = {10.1063/5.0012822},
volume = {30},
year = {2020},
}
@article{8693,
abstract = {We review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.},
author = {Chierchia, Luigi and Koudjinan, Edmond},
journal = {Regular and Chaotic Dynamics},
pages = {583–606},
publisher = {Springer},
title = {{V. I. Arnold’s “pointwise” KAM theorem}},
doi = {10.1134/S1560354719060017},
volume = {24},
year = {2019},
}