Dynamics of a Lotka–Volterra map

Volume 191 / 2006

Francisco Balibrea, Juan Luis García Guirao, Marek Lampart, Jaume Llibre Fundamenta Mathematicae 191 (2006), 265-279 MSC: Primary 58F13. DOI: 10.4064/fm191-3-5


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.


  • Francisco BalibreaDepartamento de Matemáticas
    Universidad de Murcia
    30100 Murcia (Región de Murcia), Spain
  • Juan Luis García GuiraoDepartamento de Matemática
    Aplicada y Estadística
    Universidad Politécnica de Cartagena
    C/ Paseo Alfonso XIII
    30203 Cartagena (Región de Murcia), Spain
  • Marek LampartMathematical Institute at Opava
    Silesian University at Opava
    Na Rybníčku 1 1
    746 01 Opava, Czech Republic
  • Jaume LlibreDepartament de Matemátiques
    Universitat Autónoma de Barcelona
    Bellaterra, 08193 Barcelona (Catalunya), Spain

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