Symmetry reduction by lifting for maps

Dullin, Holger R.; Lomelí, Héctor E.; Meiss, James D.

doi:10.1088/0951-7715/25/6/1709

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…But the right-hand side is zero for a 1 = 0 and a 1 = h/q and these are the candidates for singularities. Near a 1 = 0 from (13) we see that ρ ∼ c a p/2 1 where c > 0 is a constant. Thus the surface of revolution is only smooth at a 1 = 0 if p = 1.…”

Section: Case P >mentioning

confidence: 80%

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Singular reduction of resonant Hamiltonians

2018

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We investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point.

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Section: Case P >mentioning

confidence: 80%

“…With this we show that the orbit space O = N /S is a symplectic orbifold by viewing cross sections to the flow as symplectic charts on O. Related recent work is presented by Dullin et al [13]. These authors focus on diffeomorphisms and introduce the concept of reduction by lifting.…”

Section: Introductionmentioning

confidence: 80%

Singular reduction of resonant Hamiltonians

2018

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“…There is an interesting case where a close construction is exploited in other circumstances [11]. This concerns the case when a map f : M → M acts on a smooth manifold with the and it is assumed in addition that there is a symmetry for f , that is a smooth vector field v on M such that…”

Section: Divergence-free Vector Fields and Hamiltonian Systemsmentioning

confidence: 99%

On interrelations between divergence-free and Hamiltonian dynamics

Lerman

Yakovlev

2019

Journal of Geometry and Physics

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A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field X with a global cross-section there exist some 4-dimensional symplectic manifoldM ⊃ M and a smooth Hamilton function H :M → R such that for some c ∈ R one gets M = {H = c} and the Hamiltonian vector field X H restricted on this level coincides with X. For divergence free vector fields with singular points such the extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M . We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such the linearization is possible for tori with Diophantine rotation numbers.

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A development scenario connecting the ternary symmetric horseshoe with the binary horseshoe

González

Jung

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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It is explained in which way the ternary symmetric horseshoe can be obtained along a development scenario starting with a binary horseshoe. We explain the case of a complete ternary horseshoe in all detail and then give briefly some further incomplete cases. The key idea is to start with a three degrees of freedom system with a rotational symmetry, reduce the system with the help of the conserved angular momentum to one with two degrees of freedom where the value of the conserved angular momentum acts as a parameter and then let its value go to zero.

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Symmetry reduction by lifting for maps

Cited by 3 publications

References 31 publications

Singular reduction of resonant Hamiltonians

Singular reduction of resonant Hamiltonians

On interrelations between divergence-free and Hamiltonian dynamics

A development scenario connecting the ternary symmetric horseshoe with the binary horseshoe

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