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2013
Impact Factor

Ivan Ovsyannikov

10, Ulyanova st. 603005, Nizhny Novgorod, Russia
Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University

Publications:

Gonchenko S. V., Gordeeva O. V., Lukyanov V. I., Ovsyannikov I. I.
On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node
2014, vol. 19, no. 4, pp.  461-473
Abstract
We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
Keywords: saddle-node, homoclinic tangency, Arnold tongues
Citation: Gonchenko S. V., Gordeeva O. V., Lukyanov V. I., Ovsyannikov I. I.,  On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 461-473
DOI:10.1134/S1560354714040029
Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.
Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points
2014, vol. 19, no. 4, pp.  495-505
Abstract
It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.
Keywords: Homoclinic tangency, rescaling, 3D Hénon map, bifurcation, Lorenz-like attractor
Citation: Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.,  Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 495-505
DOI:10.1134/S1560354714040054
Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.
Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation
2006, vol. 11, no. 2, pp.  191-212
Abstract
We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i\varphi}, \lambda e^{-i\varphi}, \gamma)$, where $0<\lambda<1<|\gamma|$ and $|\lambda^2 \gamma|=1$. We show that in a three-parameter family, $g_\varepsilon$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\varepsilon=0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a ''wild-hyperbolic'' Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian
Keywords: saddle-focus fixed point, three-dimensional quadratic map, homoclinic bifurcation, strange attractor
Citation: Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.,  Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 191-212
DOI: 10.1070/RD2006v011n02ABEH000345

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