Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom

Christov, Ognyan

doi:10.1007/s10569-011-9389-4

Cited by 25 publications

(31 citation statements)

References 26 publications

(24 reference statements)

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“…Remark 1 We expect that the three 'resonant terms' of order 3 make the normal form truncated at order 3 non-integrable, similar to the (definite) 1:1:2 resonance for which nonintegrability has been proven in the absence of extra symmetries [11]; see also [7] where the same result could be achieved for the 1:2:3 and 1:2:4 resonances.…”

Section: Introductionmentioning

confidence: 83%

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Efstathiou

Hanßmann

Marchesiello

2019

Journal of Geometry and Physics

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We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:−2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:−2 resonance. arXiv:1807.09453v1 [math.DS] 25 Jul 2018 1 2κto include the case λ = 0.

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Section: Introductionmentioning

confidence: 83%

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Efstathiou

Hanßmann

Marchesiello

2019

Journal of Geometry and Physics

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“…The cases where both resonances (9) are of order 3 are called of genuine first order [39], these are the cases 1 : ±2 : ±3 and 1 : ±2 : ±4. Here already the cubic normal form is expected to be non-integrable, in the definite cases 1 : 2 : 3 and 1 : 2 : 4 this has been proven in [12].…”

Section: Non-integrable Resonancesmentioning

confidence: 83%

“…In the definite case 1 : 1 : 2 the cubic normal form is known to be non-integrable [12,18] while for m = ±1 : 2 : 2 the cubic normal form may serve as intermediate system [1,25,31]; in the additional cases of genuine second order the cubic normal form is again trivial. In indefinite cases with double eigenvalues of opposite symplectic sign these may be involved in a Krein collision and leave the imaginary axis during a subordinate Hamiltonian Hopf bifurcation.…”

Section: Non-integrable Resonancesmentioning

confidence: 99%

Perturbations of Superintegrable Systems

Hanβmann

2015

Acta Appl Math

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A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d − n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n + 1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis.

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“…Exactly as in [24], we speak of a genuine first-order resonance if there are at least two linearly independent vectors k andk with |k| = 3 and |k| = 3, such that k.ω =k.ω = 0 (also see [6,21]). Notice that by [21], if the annihilators with bounded norm by µ+2, span a codimension 1 sublattice of Z 4 , and ν ∈ N is minimal, then ω defines a genuine νth-order resonance.…”

mentioning

confidence: 99%

“…A general discussion on the integrability of the normal form of Hamiltonians can be found in [26] and also an algebraic proof for the non-integrability of first-order resonances is found in [6].…”

mentioning

confidence: 99%

The coupled 1:2 resonance in a symmetric case and parametric amplification model

Mazrooei‐Sebdani¹,

Yousefi²

2021

DCDS-B

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This paper deals with the coupled Hamiltonian 1:2 resonance, i.e. the Hamiltonian 1:2:1:2 resonance. This resonance is of the first order. We isolate several integrable cases. Our main focus is on two models. In the first part of the paper, we present a discrete symmetric normal form truncated to order three and we compute the relative equilibria for its corresponding system. In the second part, the paper is devoted to the study of the Hamiltonian describing the four-wave mixing (FWM) model. In addition to the Hamiltonian, the corresponding system possesses three more independent integrals. We use these integrals to obtain estimates for the phase space and total energy. Further, we compute the relative equilibria of the FWM system for the 1:2:1:2 resonance. Finally, we carry out some numerical experiments for the detuned system.

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Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom

Cited by 25 publications

References 26 publications

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

Perturbations of Superintegrable Systems

The coupled 1:2 resonance in a symmetric case and parametric amplification model

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