On homoclinic attractors of three-dimensional flows

Abstract

The main goal is to construct a classification of such attractors and to distinguish among them the classes of pseudohyperbolic attractors which chaotic dynamics is preserved under perturbations of the system. The main research method is a qualitative method of saddle charts, which consists of constructing an extended bifurcation diagram on the plane of the system parameters in the form $\dot x=y+g_1(x,y,z), \dot y=z+g_2(x,y,z), \dot z=Ax+By+Cz+g_3(x,y,z), \;\; g_i(0,0,0) = (g_i)^\prime_x(0,0,0) = (g_i)^\prime_y(0,0,0) = (g_i)^\prime_z(0,0,0) = 0, \; i = 1, 2, 3$, the linearization matrix of which is represented in the Frobenius form, and the eigenvalues that determine the type of equilibrium state are expressed only through the coefficients A, B, C. The pseudohyperbolicity of the attractors under consideration is verified by means of a numerical method which helps to check the continuity of the subspaces of strong contractions and volume expansion on the attractor. The homoclinic nature of attractors is established using the numerical method of calculating the distance from an attractor to a saddle equilibrium. Results. An extended bifurcation diagram is constructed on the parameter plane (A,B), on which the stability region of the equilibrium state is highlighted, as well as six regions corresponding to two different types of spiral figure-eight attractors, Shilnikov attractor, Lorenz-like attractor, Shilnikov saddle attractor, and Lyubimov–Zaks–Rovella attractor. The pseudohyperbolicity of the Lorenz-like attractor is confirmed numerically. For the attractors of Lyubimov– Zaks–Rovella, it is shown that despite the continuity of strong contracting and volume-expanding subspaces such attractors cannot be pseudohyperbolic. The paper discusses that in three-dimensional flows, in addition to Lorenz-like attractors, only Shilnikov saddle attractors containing a saddle equilibrium state with a two-dimensional unstable manifold can be pseudohyperbolic. However, we currently do not know examples of such attractors.

Keywords

• chaotic attractor
• saddle equilibrium
• saddle-focus
• homoclinic trajectory
• pseudohyperbolicity
• lorenz attractor
• spiral attractor
• shilnikov attractor