Bifurcations and monodromy of the axially symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e4217" altimg="si101.svg"><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> resonance

Efstathiou, Konstantinos; Hanßmann, Heinz; Marchesiello, Antonella

doi:10.1016/j.geomphys.2019.103493

Cited by 5 publications

(3 citation statements)

References 21 publications

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“…We stress again that we always consider δ = 0, because adding the additional quadratic term in L 3 to the Hamiltonian deforms the bifurcation diagram, but does not essentially change it. Bifurcations similar to those found here have recently been described in [34,17], in particular also the related quantum monodromy in [34]. Let us start with the ordinary Lagrange top, c 2 = 0 [8].…”

Section: Bifurcation Diagramsupporting

confidence: 79%

The Harmonic Lagrange Top and the Confluent Heun Equation

Dawson,

Dullin,

Nguyen

2021

Preprint

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The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on T * S 3 . This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.

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Section: Bifurcation Diagramsupporting

confidence: 79%

The Harmonic Lagrange Top and the Confluent Heun Equation

Dawson,

Dullin,

Nguyen

2021

Preprint

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“…The approach we take to study the 2:4 resonance has become rather standard, compare with [25,12,20,24,14] and references therein. Normalizing about the periodic flow of the resonant oscillator introduces an extra continuous symmetry, cf.…”

Section: What Is Newmentioning

confidence: 99%

“…Remark 4.3 The simple geometry of the parabola allows to immediately conclude stability or instability of the equilibria and how these come into existence through centre-saddle and Hamiltonian flip bifurcations. The corresponding formulas may as well be searched for as double roots of the difference of the polynomials describing phase space and energy level set [14]. For more involved expressions than the present cubic, which is well approximated by a parabola, an algebraic point of view can support the present geometric approach, relying on the resultant of two polynomials and related tools.…”

Section: Regular Equilibriamentioning

confidence: 99%

On the detuned 2:4 resonance

Hanssmann,

Marchesiello,

Pucacco

2020

Preprint

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We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -and the normal form -both have no cubic terms. This Z 2 × Z 2 -symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where H = T + V would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.

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Reduction and reconstruction procedure of a family of polynomial Hamiltonians in 1:1:2 and 1:1:-2 resonances

Pérez Rothen,

Vidal

2024

Dynamical Systems

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

Cited by 5 publications

References 21 publications

The Harmonic Lagrange Top and the Confluent Heun Equation

The Harmonic Lagrange Top and the Confluent Heun Equation

On the detuned 2:4 resonance

Reduction and reconstruction procedure of a family of polynomial Hamiltonians in 1:1:2 and 1:1:-2 resonances

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