On Some Types of Homoclinic Attractors of Three-dimensional Maps and the Mechanisms of their Occurrence

In this paper, we conducted a study of discrete homoclinic attractors. Their study is very important in the theory of dynamical systems, because they can be wild pseudo-hyperbolic attractors, that is, real strange attractors that do not contain stable periodic trajectories. We classified homoclinic attractors into 4 types, depending on the value of the saddle point multipliers, and selected a real example for each of them. One of these is the discrete Lorentz attractor. In this paper, we describe in detail possible scenarios of its occurrence, and also conduct a full-fledged bifurcation analysis for fixed and periodic points for three-dimensional Eno mapping. Phase portraits are presented, which track the scenarios of the attractor birth at various parameters. A map of Lyapunov exponents and a map of modes is constructed. Next, we consider the chapter on pseudohyperbolicity. Sufficient and necessary conditions for the existence of pseudohyperbolic attractors are determined here. A numerical method for testing attractors - LNP graph-is also studied and applied.