Revista de la
Unión Matemática Argentina

Temporary Suspension of Submission and Review System

Due to technical issues, our submission and review system is temporarily out of service. Until the system is back, we will not be accepting new submissions. For any other urgent inquiries, please contact revuma@criba.edu.ar.

We apologize for the inconvenience and appreciate your understanding.

Global attractors in the parametrized Hénon–Devaney map
Bladismir Leal and Sergio Muñoz
Volume 65, no. 1 (2023), pp. 79–101    

https://doi.org/10.33044/revuma.2133

Download PDF

Abstract

Given two positive real numbers $a$ and $c$, we consider the (two-parameter) family of nonlinear mappings \[ F_{a,c}(x,y)=\left(ax+\frac{1}{y}, \, c y-\frac{c}{y}-a c x\right). \] $F_{1,1}$ is the classical Hénon–Devaney map. For a large region of parameters, we exhibit an invariant non-bounded closed set with fractal structure which is a global attractor. Our approach leads to an in-depth understanding of the Hénon–Devaney map and its perturbations.

References

  1. R. L. Devaney, The baker transformation and a mapping associated to the restricted three-body problem, Comm. Math. Phys. 80 (1981), no. 4, 465–476. MR 0628505.
  2. M. Hénon, Generating Families in the Restricted Three-Body Problem. II. Quantitative Study of Bifurcations, Lecture Notes in Physics. New Series m: Monographs, 65, Springer, Berlin, 2001. MR 1874416.
  3. B. Leal, G. Mata, and S. Muñoz, Families of transitive maps on $\mathbb{R}$ with horizontal asymptotes, Rev. Un. Mat. Argentina 59 (2018), no. 2, 375–387. MR 3900279.
  4. B. Leal and S. Muñoz, Hénon-Devaney like maps, Nonlinearity 34 (2021), no. 5, 2878–2896. MR 4260781.
  5. B. Leal and S. Muñoz, Invariant Cantor sets in the parametrized Hénon-Devaney map, Dyn. Syst. 37 (2022), no. 1, 105–126. MR 4408079.
  6. F. Lenarduzzi, Generalized Hénon-Devaney Maps of the Plane, PhD Thesis, IMPA, Rio de Janeiro, 2016.
  7. F. Lenarduzzi, Recoding the classical Hénon-Devaney map, Discrete Contin. Dyn. Syst. 40 (2020), no. 7, 4073–4092. MR 4097535.
  8. S. Muñoz, Robust transitivity of maps of the real line, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 1163–1177. MR 3277190.