PHYSICAL REVIEW E VOLUME 49, NUMBER 4 APRIL 1994 Characterization of the Lorenz system, taking into account the equivariance of the vector field C. Letellier, P. Dutertre, and G. Gouesbet 76130 Mont-Saint-Aignan, France* (Received 29 October 1993) We characterize the chaotic attractors of the Lorenz system associated with R =28 and R=90 (re- duced Rayleigh number) by using a partition that takes into account the equivariance of the vector Qeld. The population of unstable periodic orbits is extracted and encoded respectively with binary and three letter symbolic dynamics. Templates are proposed for these R values. PACS number(s): 05. 45. + b I. INTRODUCTION In the past few years several works discussed the topo- logical description of chaotic attractors. In particular, the idea has arisen that an attractor can be described by the population of periodic orbits, their related symbolic dynamics, and their linking numbers [1]. In three dimen- sional cases, periodic orbits may be viewed as knots [2] and, consequently, they are robust with respect to smooth parameter changes and allow the definition of topological invariants under isotopy. A topological analysis procedure may consist of a num- ber of steps. A population of unstable periodic orbits is first extracted from the How. Then, the topological or- ganization of the unstable periodic orbits is determined by computing interlinking and self-linking numbers. From such an analysis of a few small period orbits, a tem- plate is built. This template can be used to predict the linking numbers characterizing the orbits. The compar- ison between template predictions and topological invari- ant measurements provides a checking of the template prediction. Although the case of asymmetric systems is well docu- mented, such a topological characterization is not fully understood for equivariant systems. For instance, the Lorenz system template proposed by Mindlin et al. [1] is not consistent with the Lorenz map [3]. However, the Mindlin et al. template corresponds to another set of control parameters, after the homoclinic explosion. Indeed, this template is composed of two bands without any local torsion, thus convicting with the existence of the decreasing monotonic branch of the Lorenz map (a decreasing branch in a map must be associated with a band whose local torsion is odd [2]). To solve this con- tradiction, we propose an equivariant description of the Lorenz attractor. A binary symbolic dynamics, based on the Lorenz map, is used to encode all orbits extracted from the attractor up to period 8 (for the reduced Ray- leigh number R =28). An equivariant template is then extracted and checked. Also, a three letter equivariant dynamics is proposed for more developed chaos (R =90}, 'FAX: (33) 35 52 83 90 allowing one to encode the population of unstable orbits. The corresponding template is again extracted and checked. II. EQUIVARIANT CHARACTERIZATION OF THE LORENZ SYSTEM y= 0 — 1 0 0 0 1 (2) The Lorenz equivariant is a Z2 symmetry, i. e. , y =I. The Lorenz system remains unchanged if x is replaced by yx, i. e. , if x(t) is a solution, then yx(t ) is also a solution. Periodic orbits of the Lorenz system may be symmetric or asymmetric (degenerate in Cvitanovic terminology [4]) with respect to the Z2 symmetry. Symmetric orbits are globally invariant under the action of y. Asymmetric or- bits are mapped to their symmetric configuration and therefore appear by pairs. B. First-return maps The strange chaotic Lorenz attractor for A, =(28, 10, — ', ) is organized around three fixed points Co(x =y =z =0) and C+(x+ =+Vb(R — l), y+ =x+, z+ =R — 1) . The Lorenz map reads M„+, =g(M„), in which M„ is the nth z maximum of the time series. These maximum values may be obtained from the intersection of the tra- jectory with two hypersurfaces X+ and X defined by A. Uector Beld equivariance A vector field f (A. , x( t ) ) is equivariant if f(k, yx(t) ) =y f(A, , x(t) }, in which x(t ) is a real-valued vector, t is the time, A, is a parameter vector, f is a smooth vector-valued function, and y is a matrix defining the equivariance. The Lorenz system [3] with parameter vector A, =(R, o, b ) and variables x= (x,y, z ), and the usual no- tations, is equivariant with the equivariant matrix read- ing — 1 0 0 1063-651X/94/49(4)/3492(4)/$06. QQ 49 3492 1994 The American Physical Society