Periodic Orbits of Planar Integrable Birational Maps

Gálvez-Carrillo, Imma; Mañosa, Vı́ctor

doi:10.1007/978-3-319-12328-8_2

Springer Proceedings in Mathematics &Amp; Statistics

2015

DOI: 10.1007/978-3-319-12328-8_2

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Periodic Orbits of Planar Integrable Birational Maps

Imma Gálvez-Carrillo

Vı́ctor Mañosa

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2024

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Cited by 2 publications

References 42 publications

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Lie symmetries of birational maps preserving genus 0 fibrations

Llorens

Mañosa

2015

Journal of Mathematical Analysis and Applications

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We prove that any planar birational integrable map, which preserves a fibration given by genus 0 curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods of the considered maps.

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Lie symmetries of birational maps preserving genus 0 fibrations

Llorens

Mañosa

2015

Journal of Mathematical Analysis and Applications

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Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

Françoise,

Fournier-Prunaret

2024

Int. J. Bifurcation Chaos

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We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map [Formula: see text]. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map [Formula: see text] and of its inverse [Formula: see text]. We show that the map [Formula: see text] is the product of two involutions. The nature of the fixed points of [Formula: see text] is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping [Formula: see text] is integrable for [Formula: see text] and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for [Formula: see text].

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Periodic Orbits of Planar Integrable Birational Maps

Cited by 2 publications

References 42 publications

Lie symmetries of birational maps preserving genus 0 fibrations

Lie symmetries of birational maps preserving genus 0 fibrations

Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

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