Ivan Ovsyannikov
Publications:
Ovsyannikov I. I.
On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles
2022, vol. 27, no. 2, pp. 217231
Abstract
Lorenz attractors are important objects in the modern theory of chaos. The reason,
on the one hand, is that they are encountered in various natural applications (fluid dynamics,
mechanics, laser dynamics, etc.). On the other hand, Lorenz attractors are robust in the sense
that they are generally not destroyed by small perturbations (autonomous, nonautonomous,
stochastic). This allows us to be sure that the object observed in the experiment is exactly a
chaotic attractor rather than a longtime periodic orbit.
Discretetime analogs of the Lorenz attractor possess even more complicated structure — they allow homoclinic tangencies of invariant manifolds within the attractor. Thus, discrete Lorenz attractors belong to the class of wild chaotic attractors. These attractors can be born in codimensionthree local and certain global (homoclinic and heteroclinic) bifurcations. While various homoclinic bifurcations leading to such attractors have been studied, for heteroclinic cycles only cases where at least one of the fixed points is a saddlefocus have been considered to date. In the present paper the case of a heteroclinic cycle consisting of saddle fixed points with a quadratic tangency of invariant manifolds is considered. It is shown that, in order to have threedimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lowerdimensional global invariant manifolds. Thus, it is assumed that either the quadratic tangency or the transversal heteroclinic orbit is nonsimple. The main result of the paper is the proof that the original system is the limiting point in the space of dynamical systems of a sequence of domains in which the diffeomorphism possesses discrete Lorenz attractors. 
Gonchenko S. V., Gordeeva O. V., Lukyanov V. I., Ovsyannikov I. I.
On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddlenode
2014, vol. 19, no. 4, pp. 461473
Abstract
We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddlenode fixed point. On a parameter plane we build a bifurcation diagram for singleround periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the wellknown codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddlenode fixed point with a transversal homoclinic orbit is discussed.

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. 495505
Abstract
It was established in [1] that bifurcations of threedimensional diffeomorphisms with a homoclinic tangency to a saddlefocus fixed point with the Jacobian equal to 1 can lead to Lorenzlike strange attractors. In the present paper we prove an analogous result for threedimensional 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.

Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.
Chaotic dynamics of threedimensional Hénon maps that originate from a homoclinic bifurcation
2006, vol. 11, no. 2, pp. 191212
Abstract
We study bifurcations of a threedimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddlefocus 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 threeparameter 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 threedimensional Hénonlike map. This map possesses, in some parameter regions, a ''wildhyperbolic'' Lorenztype strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these threedimensional Hénon maps occupy in the class of threedimensional quadratic maps with constant Jacobian
