Geodesic
Dynamics of a Lotka–Volterra map
Francisco Balibrea 1   ; Juan Luis García Guirao 2   ; Marek Lampart 3   ; Jaume Llibre 4
1 Departamento de Matemáticas Universidad de Murcia 30100 Murcia (Región de Murcia), Spain
2 Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena C/ Paseo Alfonso XIII 30203 Cartagena (Región de Murcia), Spain
3 Mathematical Institute at Opava Silesian University at Opava Na Rybníčku 1 1 746 01 Opava, Czech Republic
4 Departament de Matemátiques Universitat Autónoma de Barcelona Bellaterra, 08193 Barcelona (Catalunya), Spain
Fundamenta Mathematicae, Tome 191 (2006) no. 3, pp. 265-279 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Given the plane triangle with vertices $(0,0)$, $(0,4)$ and $(4,0)$ and the transformation $F:(x,y) \mapsto (x(4-x-y),xy)$ introduced by A. N. Sharkovski\uı, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior of the Schrödinger equation.
DOI : 10.4064/fm191-3-5
Mots-clés : given plane triangle vertices transformation mapsto x y introduced sharkovski prove existence following objects unique invariant curve spiral type periodic trajectory period given explicitly periodic trajectory period described approximately decomposition triangle which helps understand global dynamics discrete system which linked behavior schr dinger equation

Francisco Balibrea  1   ; Juan Luis García Guirao  2   ; Marek Lampart  3   ; Jaume Llibre  4

1 Departamento de Matemáticas Universidad de Murcia 30100 Murcia (Región de Murcia), Spain
2 Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena C/ Paseo Alfonso XIII 30203 Cartagena (Región de Murcia), Spain
3 Mathematical Institute at Opava Silesian University at Opava Na Rybníčku 1 1 746 01 Opava, Czech Republic
4 Departament de Matemátiques Universitat Autónoma de Barcelona Bellaterra, 08193 Barcelona (Catalunya), Spain 
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     author = {Francisco Balibrea and Juan Luis Garc{\'\i}a Guirao and Marek Lampart and Jaume Llibre},
     title = {Dynamics of a {Lotka{\textendash}Volterra} map},
     journal = {Fundamenta Mathematicae},
     pages = {265--279},
     year = {2006},
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     doi = {10.4064/fm191-3-5},
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     url = {http://geodesic.mathdoc.fr/articles/10.4064/fm191-3-5/}
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Francisco Balibrea; Juan Luis García Guirao; Marek Lampart; Jaume Llibre. Dynamics of a Lotka–Volterra map. Fundamenta Mathematicae, Tome 191 (2006) no. 3, pp. 265-279. doi: 10.4064/fm191-3-5

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