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Regular version of the site
Of all publications in the section: 5
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Article
Kazakov A., Ветчанин Е. В. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2016. Vol. 26. No. 4. P. 1650063-1-1650063-13.

In this paper, we consider a system governing the motion of two point vortices in a flow excited by an external acoustic forcing. It is known that the system of two vortices is integrable in the absence of acoustic forcing. However, the addition of the acoustic forcing makes the system much more complex, and the system becomes nonintegrable and loses the phase volume preservation property. The objective of our research is to study chaotic dynamics and typical bifurcations. Numerical analysis has shown that the reversible pitchfork bifurcation is typical. Also, we show that the existence of strange attractors is not characteristic for the system under consideration

Added: Jun 10, 2016
Article
Demina M.V., Valls C. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2019.

We give the complete classi cation of irreducible invariant algebraic curves in family (I) of the Chinese classi cation. In addition, we provide a complete and correct proof of the non{existence of algebraic limit cycles for these equations.

Added: Oct 22, 2019
Article
Gonchenko S. V., Gonchenko A. S., Kazakov A. et al. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2018. Vol. 28. No. 11. P. 1830036-1-1830036-29.

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the socalled LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized H´enon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of fourdimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

Added: Oct 26, 2018
Article
Pochinka O., Grines V., Zhuzhoma E. V. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2014. Vol. 24. No. 8. P. .

In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time.

Added: Sep 11, 2014
Article
Kazakov A., Gonchenko A. S., Gonchenko S. V. et al. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2014. Vol. 24. No. 8. P. 1440005-1440030.

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincar´ e maps in models of nonholonomic mechanics

Added: Mar 29, 2015