KAM theory:  The legacy of Kolmogorov's 1954 paper

Broer, Hendrik

doi:10.1090/s0273-0979-04-01009-2

Cited by 49 publications

(34 citation statements)

References 66 publications

(75 reference statements)

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“…Observe that c k 1 k 2 k 3 k 4 is invariant under the transpositions k 1 ↔ k 2 and k 3 ↔ k 4 . Hence (52) follows once we prove that…”

Section: In Particular If N Is Odd Then S Kmentioning

confidence: 99%

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Results on Normal Forms for FPU Chains

Henrici

Kappeler

2007

Commun. Math. Phys.

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Abstract:In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point.

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“…Observe that c k 1 k 2 k 3 k 4 is invariant under the transpositions k 1 ↔ k 2 and k 3 ↔ k 4 . Hence (52) follows once we prove that…”

Section: In Particular If N Is Odd Then S Kmentioning

confidence: 99%

“…where the quadruple (i, j, k, m) is a permutation of (1,2,3,4) and Proof. By a straightforward computation one verifies that the sets of the form (79) or (80) satisfy (71).…”

Section: A Proof Of Lemma 43mentioning

confidence: 99%

Results on Normal Forms for FPU Chains

Henrici

Kappeler

2007

Commun. Math. Phys.

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“…For overviews see [12,49,91]. The quasi-periodic bifurcations are inspired by the classical ones in which equilibria or periodic orbits are replaced by quasi-periodic tori.…”

Section: 'Next Cases'mentioning

confidence: 99%

Resonance and Singularities

Broer

Vegter

2013

Springer Proceedings in Mathematics &Amp; Statistics

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The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples.

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“…Instead of the original system in (z, t) we pull the Fig. 8 Orbits of the Poincaré mapping of the swing (12) system back along , so obtaining a Z q -equivariant system on the (ζ, t)-covering space.…”

Section: Parametric Resonancementioning

confidence: 99%

“…For overviews see [12,48,91]. The quasi-periodic bifurcations are inspired by the classical ones in which equilibria or periodic orbits are replaced by quasi-periodic tori.…”

Section: 'Next Cases'mentioning

confidence: 99%