ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Pankratova I. N. Hopf bifurcations of cycles of period two of two-dimensional logistic map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 5, pp. 53-66. DOI: 10.18500/0869-6632-2008-16-5-53-66

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 163)
Article type: 
517.9, 517.53

Hopf bifurcations of cycles of period two of two-dimensional logistic map

Pankratova Irina Nikolaevna, The Republican State Enterprise "Institute of Mathematics of the Ministry of Education and Science of the Republic of Kazakhstan"

Maps having cycles of period two in which Hopf bifurcations of new cycles occur are localized in the family of two-dimensional logistic maps. For the purposes of illustration of the bifurcation property one-dimensional sections of bifurcation diagrams with one fixed parameter for two-parameters’ first-return maps of two-dimensional logistic maps are given.

Key words: 
  1. Pankratova IN. On limit sets of a multidimensional analogue of a nonlinear logistic difference equation. Differential Equations. 1996;32(7):995–997 (in Russian).
  2. Sharkovsky AN, Kolyada SF, Sivak AG, Fedorenko VV. Dynamics of One-Dimensional Mappings. Kiev: Naukova Dumka; 1989. 216 p. (in Russian).
  3. Feigenbaum M. Universality in the behavior of nonlinear systems. Sov. Phys. Usp. 1983;141(2):343–374 (in Russian). DOI: 10.3367/UFNr.0141.198310e.0343.
  4. Poston T, Stewart I. Catastrophe Theory And Its Applications. Dover Publications; 1996. 512 p.
  5. Pankratova IN. Representation of many-group population model as one-species population model with many parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(6):135–142 (in Russian). DOI: 10.18500/0869-6632-2005-13-5-135-142.
  6. Leslie PH. On the use of matrices in certain population mathematics. Biometrika. 1945;33(3):183–212. DOI: 10.2307/2332297.
  7. Svirezhev YM, Logofet DO. Stability of Biological Populations. Moscow: Nauka; 1978. 352 p. (in Russian).
  8. Caswell H. Matrix Population Models: Construction, Analysis And Interpretation. Sunderland, Massachusettes, USA: Sunauer Associates Inc.; 1989. 722 p.
  9. Logofet DO. Once again on the nonlinear Leslie model: asymptotic behavior of trajectories in primitive and imprimitive cases. Dokl. Math. 1991;43(3):861–865. 
  10. Gantmacher FR. Matrix Theory. In 2 Volumes. Chelsea; 1960.
Short text (in English):
(downloads: 90)