IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 41 (2008) 485101 (13pp) doi:10.1088/1751-8113/41/48/485101 On the topology of the L ¨ u attractor and related systems S Anastassiou 1 , T Bountis 1 and Y G Petalas 2 1 Center for Research and Applications of Nonlinear Systems (CRANS), Department of Mathematics, University of Patras, GR-26500, Patras, Greece 2 Computational Intelligence Laboratory (CI Lab), Artificial Intelligence Research Center (UPAIRC), Department of Mathematics, University of Patras, GR-26500, Patras, Greece E-mail: SAnastassiou@gmail.com Received 27 June 2008, in final form 11 September 2008 Published 22 October 2008 Online at stacks.iop.org/JPhysA/41/485101 Abstract We use well-established methods of knot theory to study the topological structure of the set of periodic orbits of the L¨ u attractor. We show that, for a specific set of parameters, the L ¨ u 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 L¨ u and Lorenz attractors, claimed, for these parameter values, by some authors on the basis of non-topological observations. However, we show that the L¨ u 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 L¨ u kind with higher order symmetry is constructed and some remarks on the Chen attractor are also presented. PACS number: 05.45.−a 1. Introduction L¨ u and Chen introduced in [11] a simple 3-parameter family of ordinary differential equations (ODEs), nowadays called the L¨ u system, which exhibits chaotic behaviour. The equations of this system are ˙ x = a(y − x) ˙ y =−xz + cy ˙ z = xy − bz (1) where a,b,c are real parameters. In the same paper, L¨ u and Chen integrated these ODEs numerically, for fixed a = 36 and b = 3, varying only the third parameter. They observed that, when 12.7 <c< 17, the attractor generated by (1) is pictorially ‘similar’ to the well-known 1751-8113/08/485101+13$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1