On hyperbolic attractors in a modified complex Shimizu–Morioka system

Abstract

We present a modified complex-valued Shimizu–Morioka system with a uniformly hyperbolic attractor. We show that the numerically observed attractor in the Poincaré cross section expands three times in the angular direction and strongly contracts in the transversal directions, similar in structure to the Smale–Williams solenoid. This is the first example of a modification of a system with a genuine Lorenz attractor, but manifesting a uniformly hyperbolic attractor instead. We perform numerical tests to show the transversality of tangent subspaces, a pivotal property of uniformly hyperbolic attractors, for both the flow system and its Poincaré map. We also observe that no genuine Lorenz-like attractors appear in the modified system.