Ivan Ovsyannikov

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 long-time periodic orbit.
Discrete-time 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 codimension-three 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 saddle-focus 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 three-dimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lower-dimensional 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.

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.

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.

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