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Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps. (English) Zbl 1376.37047
Summary: 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 positive maximal Lyapunov exponent 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 Hénon maps of the form $$\overline{x} = y$$, $$\overline{y} = z$$, $$\overline{z} = B x + A z + C y + g(y, z)$$, where $$A$$, $$B$$ and $$C$$ are parameters ($$B$$ is the Jacobian) and $$g(0, 0) = g^\prime(0, 0) = 0$$.

MSC:
 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 39A33 Chaotic behavior of solutions of difference equations
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