On the classification of homoclinic attractors of three-dimensional flows

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz-like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.