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Article

О классификации гомоклинических аттракторов трехмерных потоков

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors, which goes back to the papers S.V. Gonchenko, D.V. Turaev A.L. Shilnikov and L.P. Shilnikov, is proposed. By homoclinic we mean strange attractors containing a specific saddle equilibrium together with its unstable manifold. Moreover, the type of such an attractor is determined by the eigenvalues of the equilibrium. The classification of homoclinic attractors is based on a bifurcation analysis of systems of the form $\dot x=y+g_1(x,y,z), \dot y=z+g_2(x,y,z), \dot z=Ax+By+Cz+g_3(x,y,z), \;\; g_i(0,0,0) = (g_i)^\prime_x(0,0,0) = (g_i)^\prime_y(0,0,0) = (g_i)^\prime_z(0,0,0) = 0, \; i = 1, 2, 3$, whose linearization matrix is represented in the Frobenius form, and the eigenvalues are determined by the coefficients $A, B$ and $C$. In the parameters space $A, B$ and $C$, an extended bifurcation diagram is constructed, where 7 regions corresponding to attractors of various types are distinguished. It is noted that a wide class of three-dimensional flows can be reduced to the class of systems under consideration. The paper also discusses problems related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows (stability of chaotic dynamics to changes of system parameters). It is proved that in three-dimensional flows only two types of homoclinic attractors can be pseudo-hyperbolic: Lorenz-like attractors containing a saddle equilibrium state with a two-dimensional stable manifold whose saddle value is positive; as well as Shilnikov saddle attractors containing a saddle equilibrium with a two-dimensional unstable manifold.