A proof of the isoenergetic KAM-theorem from the “ordinary” one

Broer, Hendrik; Huitema, George

doi:10.1016/0022-0396(91)90160-b

Cited by 36 publications

(28 citation statements)

References 6 publications

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“…The paper [33] gives a complete proof of the full KAM theorem and establishes, in particular, the C ∞ -dependence of the KAM tori on the frequencies, in the sense of Whitney. The statement of Theorem 7.5 above can be extracted from the references [5] (see, in particular, Lemma 2.3 of that paper), and especially, [8]. The latter paper derives Theorem 7.5 from the ordinary KAM theorem of [33].…”

Section: Perturbations Of Completely Integrable Systemsmentioning

confidence: 99%

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Diophantine tori and spectral asymptotics for nonselfadjoint operators

Hitrik¹,

Sjöstrand²,

Ngọc³

2007

ajm

176

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We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength ǫ of the perturbation is O(h δ ) for some δ > 0 and is bounded from below by a fixed positive power of h. In the second case, ǫ is assumed to be sufficiently small but independent of h, and we describe the eigenvalues completely in a fixed h-independent domain in the complex spectral plane.1

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Section: Perturbations Of Completely Integrable Systemsmentioning

confidence: 99%

“…A more refined version of the isoenergetic KAM theorem, which also follows from the arguments of [8], gives Cantor families of Diophantine invariant tori with prescribed rotation numbers, in energy surfaces p −1 λ (E), when E ∈ neigh(0, R), with an analytic dependence on E.…”

Section: Perturbations Of Completely Integrable Systemsmentioning

confidence: 99%

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Hitrik¹,

Sjöstrand²,

Ngọc³

2007

ajm

176

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“…: ω n (I)) to the (n − 1)-dimensional projective space should have maximal rank. Compare with [6], [18], [15]. We remark that the conclusion drawn in the example of the two coupled oscillators of the previous section also would follow as an application of the iso-energetic kam theorem.…”

Section: Conservative Kam Theorymentioning

confidence: 97%

KAM theory: The legacy of Kolmogorov's 1954 paper

Broer

2004

Bull. Amer. Math. Soc.

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“…To obtain information on the tori that are generated by the bifurcation we need non-linear terms from (8). After a normalization with respect to (9), cf.…”

Section: Pitchfork Bifurcationsmentioning

confidence: 99%

“…see [20,8,10]. The resulting Cantor dust parametrising the persistent invariant tori again has relative measure meas(…”

Section: Introductionmentioning

confidence: 99%

Quasi-periodic bifurcations in reversible systems

Hanßmann

2010

Regul. Chaot. Dyn.

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Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parametrised by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behaviour of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.

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A proof of the isoenergetic KAM-theorem from the “ordinary” one

Cited by 36 publications

References 6 publications

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Diophantine tori and spectral asymptotics for nonselfadjoint operators

KAM theory: The legacy of Kolmogorov's 1954 paper

Quasi-periodic bifurcations in reversible systems

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