Variety of strange pseudohyperbolic attractors in three-dimensional generalized H'enon maps

Abstract

In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has a positive maximal Lyapunov exponents and this property is robust, i.e. it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized H\'enon maps of form $\bar x = y, \; \bar y = z, \; \bar z = Bx + Az + Cy + g(y,z)$, where $A,B$ and $C$ are parameters ($B$ is the Jacobian) and $g(0,0) = g^\prime(0,0) =0$.