On the topology of the Lü attractor and related systems

Abstract We use well-established methods of knot theory to study the topological structure of the set of periodic orbits of the Lü attractor. We show that, for a specific set of parameters, the Lü attractor is topologically different from the classical Lorenz attractor, whose dynamics is formed by a double cover of the simple horseshoe. This argues against the ‘similarity’ between the Lü and Lorenz attractors, claimed, for these parameter values, by some authors on the basis of non-topological observations. However, we show that the Lü system belongs to the Lorenz-like family, since by changing the values of the parameters, the behaviour of the system follows the behaviour of all members of this family. An attractor of the Lü kind with higher order symmetry is constructed and some remarks on the Chen attractor are also presented.