Monodromy of Hamiltonian systems with complexity 1 torus actions

Journal of Geometry and Physics

Hanßmann

Marchesiello

2019

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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.

Section: Case δmentioning

confidence: 99%

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

Journal of Geometry and Physics

Hanßmann

Marchesiello

2019

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“…We note that the present work is closely related to the works [29,39], which demonstrate how one can compute monodromy by focusing on the circle action and without using Morse theory. However, the idea of computing monodromy through energy hyper-surfaces and their Chern numbers can also be applied when we do not have a detailed knowledge of the singularities of the system; see Remark 3.5.…”

Section: Introductionmentioning

confidence: 65%

“…For this reason, it cannot be directly generalized to systems with many degrees of freedom. An approach that admits such a generalization was developed in [29,39]; we shall recall it in the next section.…”

Section: Remark 35 (Generalization)mentioning

confidence: 99%

“…We discuss in detail two examples and make a connection to the Duistermaat-Heckman theorem [21]. In Section 4 we revisit the symmetry approach to monodromy presented in the works [29,39], and link it to the rotation number [14]. The paper is concluded with a discussion in Section 5.…”

Section: Introductionmentioning

confidence: 99%

“…[9, §4.3.2],[29]) Assume that the energy-momentum map F is proper and invariant under a Hamiltonian circle action. Let γ ⊂ image(F ) be a simple closed curve in the set of the regular values of the map F .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Hamiltonian Monodromy and Morse Theory

Martynchuk

Broer

2019

Commun. Math. Phys.

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We show that Hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying Morse theory to the Hamiltonian of the system. Our proof is based on Takens's index theorem, which specifies how the energy-h Chern number changes when h passes a non-degenerate critical value, and a choice of admissible cycles in Fomenko-Zieschang theory. Connections of our result to some of the existing approaches to monodromy are discussed.

Parallel Transport Along Seifert Manifolds and Fractional Monodromy

Martynchuk

2017

Commun. Math. Phys.

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Abstract:The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard ('integer') monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.