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

For citation:

Kuznetsov A. P., Savin A. V., Sedova Y. V. Bogdanov–Takens bifurcation: from flows to discrete systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 6, pp. 139-158. DOI: 10.18500/0869-6632-2009-17-6-139-158

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Language: 
Russian
Heading: 
Methodical Papers on Nonlinear Dynamics
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2009-17-6-139-158

Bogdanov–Takens bifurcation: from flows to discrete systems

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Savin Aleksej Vladimirovich, Saratov State University
Sedova Yuliya Viktorovna, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The methodically important bifurcation – Bogdanov–Takens bifurcation – is discussed. For the primary model its bifurcations and evolution of phase portraits are described. The examples of nonlinear systems with such bifurcation are presented. The method of discrete models of construction that is founded on semi-explicit Euler scheme is discussed. On the base of the continuous prototype the discrete model of Bogdanov– Takens oscillator is constructed. The analytical analysis of bifurcations of a codimension one and two for discrete model is realized. With the help of method of charts of dynamical regimes the picture of synchronization tongues has been revealed and scaling has been demonstrated. The illustrations of destruction and disappearance of an invariant curve are given. One more map suitable for educational purposes – Bogdanov map is discussed. Some Internet resources interesting from methodically point of view are presented. 

Key words: 
Bogdanov–Takens bifurcation
discrete models
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Received: 
16.01.2009
Accepted: 
01.04.2009
Published: 
31.12.2009
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