On homoclinic attractors in three-dimension systems with constant divergency

Title On homoclinic attractors in three-dimension systems with constant divergency A. O. Kazakov1, A. D. Kozlov1, 2, A. G. Korotkov21National Research University Higher School of Economics 2Lobachevsky State University of Nizhni Novgorod – National Research University In this paper we focus on the problem of existence of homoclinic attractors in three-dimensional flows $\dot x = y, \dot y = z, \dot z = Ax+By+Cz+g(x,y), g(0,0) = g'_x(0,0) = g'_y(0,0) = 0$. Homoclinic attractors are the strange attractors which contain only one (saddle) equilibrium point. The type of such attractors is defined by eigenvalues of the equilibrium point, which depend only on parameters $A, B$, and $C$. A method of saddle charts (two-parameter diagram in which regions with different eigenvalues are drawn with different colors) together with methods of charts of maximal Lyapunov exponent and charts of the distance between an attractor and a saddle point (to verify that a saddle point belongs to the attractor) are used for searching and classifying of homoclinic attractors. Using these methods we found attractors of spiral and also Shilnikov types. Strange attractor, homoclinic trajectory, spiral chaos Kazakov A. O., Kozlov A. D., Korotkov A. G. ''On homoclinic attractors in three-dimension systems with constant divergency'' [Electronic resource]. Proceedings of the XIII International scientific conference ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 364-369. Available at: http://conf.svmo.ru/files/deamm2017/papers/paper51.pdf. - Date of access: 24.06.2019.