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).
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Language: 
Russian
Article type: 
Article
UDC: 
517.9, 517.53

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

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

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.

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Reference: 
  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.
Received: 
29.11.2007
Accepted: 
24.04.2008
Published: 
31.12.2008
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