Non-Strange Chaotic Attractors Equivalent to Their Templates
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Non-strange chaotic attractors equivalent to their templates John Starrett Department of Mathematics New Mexico Institute of Mining and Technology 801 Leroy Place, Socorro NM, 87801 USA We construct systems of three autonomous ¯rst order di®erential equations with bounded two- dimensional attracting sets M. The flows on M are chaotic and have one dimensional Poincar¶e sections whose Poincar¶emaps are di®eomorphic to maps of the interval. The attractors are two dimensional rather than fractal, and when \unzipped" they are topologically equivalent to the templates of suspended horseshoes. INTRODUCTION The analysis of the dynamics of chaotic systems can be broken roughly into two parts { the quantitative, which emphasizes measures such as Lyapunov exponents, frac- tal dimensions and power spectra [1], and the topological, where qualitative characterizations such as templates and bounding tori are considered [2]. When we think of the in¯nite set of unstable periodic orbits in a strange at- tractor as topological knots, the topology of the attrac- tor can be coded in the set of links the dynamics allows. The primary tool is a template, a construction due to Birman and Williams [3] [4], a branched two-manifold that results when the attractor is collapsed along its sta- FIG. 1: On the left, a portion of a chaotic orbit of the Rossler ble manifolds. The template supports a semi-flow which system, on the right, a template for the Rossler system with counterclockwise flow corresponding to counterclockwise flow contains the same periodic orbits as the original system, about the z axis in the attractor. with at most two additional extraneous orbits. Extracting a template from a strange attractor is a relatively straightforward, but inexact process. In the ical system whose attractor is one-dimensional. Such an simplest cases, one can guess the form of the template attractor cannot be called \strange" because it does not by making a visual inspection of the attractor and its have a fractal structure. The two-dimensional maps we Poincar¶esections to try to determine how the stretch- suspend are di®eomorphic to one-dimensional maps. Be- ing and folding is taking place, see Fig. (1). Once a cause the exact Poincar¶emap dynamics are known, such template structure is proposed, the knots and links that continuous chaotic systems might be useful test beds for are supported by that structure can be determined. One understanding the behavior of map based and continuous then attempts to verify that the periodic orbits corre- chaos control methods, and other areas where the rela- sponding to these topological knots and links exist in the tion of a Poincar¶emap to its flow might be useful. Also, dynamics of the original system. This veri¯cation pro- these attractors are the in¯nite dissipation limiting case cess is the source of the inexactness: one never knows of families of strange attractors. if one has checked a su±ciently large set of orbits. For more complicated systems, one can extract a set of peri- odic orbits by the method of close returns or by Pyragas SUSPENDING MAPS TO FLOWS [5] or similar chaos control methods [6], and determine a set of topological invariants such as linking numbers The ¯rst detailed exposition of the conditions that and relative rotation rates for these orbits. One can then must hold for an explicit suspension of a map to a flow construct a template based on symbolic dynamics given were given by Channell [8]. Mayer-Kress and Haken [9] to the orbits [7]. suspended a H¶enonmap to a cylinder R2 £ I, but the Because template construction can sometimes be a dif- resulting di®erential equations could not be integrated ¯cult iterative process, it is an interesting exercise to con- forward normally: the integrator must reset t whenever struct an explicit chaotic dynamical system in the form t = 1 and load in the ¯nal conditions at t = 1 as the of a system of three autonomous di®erential equations new initial conditions each time. Recently Nicholas[10] whose dynamics actually take place on a template. Our described a method for making an exact autonomous approach is to suspend a two-dimensional chaotic dynam- suspension of a map in an open half plane to a flow. 2 by · ¸ · ¸ x^ ¡F n+1 = 2 ; (4) z^n+1 F1 then the result is rotated by ¼=2 clockwise · ¸ · ¼ ¼ ¸ · ¸ · ¸ xn+1 cos( 2 ) sin( 2 ) ¡F2 F1 = ¼ ¼ = : (5) zn+1 ¡ sin( ) cos( ) F1 F2 FIG. 2: Left to right, the attractors of the quadratic map, 2 2 logistic map and cubic map, in black, along with the rotated We suspend the maps to flows by continuous interpola- functions along which the attractors lie, in gray. tions between initial and ¯nal conditions. The interpola- tion is performed using the basic interpolating functions C = cos( ¼ t) and S = sin( ¼ t), which also appear in a 2 · 2 ¸ The construction embeds a flow on a quotient space CS Q = R2 £ [0; 1]= », where 0 and 1 are identi¯ed in rotation matrix R = . As interpolating di®eo- 3 ¡SC [0; 1], in R by wrapping it around the z axis. More morphisms, these are exactly, the flow Á on Q~ = R2 £ S1 is embedded in R3 2 3 ³¼ ´ by truncating R to (¡a; 1) £ R and constructing R C2 = cos2 t (6) as (0; 0; z) [F((¡a; 1) £ R), where F((¡a; 1) £ R) is ³ 2 ´ 3 ¼ a book-leaf foliation of R =(0; 0; z) rotationally symmet- S2 = sin2 t (7) ric about the z axis. The method in [10] was demon- 2 strated on two-dimensional maps with strange attractors, taking initial to ¯nal conditions by C2x +S2x as t varies a H¶enonand a Du±ng map. While this method only sus- 0 1 from 0 to 1. Because of the scaling of the argument, a pends the dynamics in the right half plane, it produces a quarter clockwise rotation and a smooth interpolation system of di®erential equations that can be integrated in between initial and ¯nal conditions is accomplished as t a normal fashion. We use this method to suspend two- goes from 0 to 1, and this suspends the map to a cylinder dimensional maps with one-dimensional attracting sets, R2 £ I. resulting in attractors with no fractal structure, but with In anticipation of our embedding in R3, we write the veri¯able chaotic dynamics. suspension to a cylinder R2 £ I as Consider the di®erence equations 2 3 2 32 2 2 3 6 x 7 6 C 0 S 76 C x0 ¡ S F2 7 6 7 6 76 7 6 7 6 76 7 2 6 7=6 76 7= xn+1 = 4=3 ¡ 5=4 xn + ®zn zn+1 = ¡xn (1) 6 y 7 6 0 1 0 76 0 7 4 5 4 54 2 2 5 z ¡S 0 C C z0 + S F1 xn+1 = 4 xn(1 ¡ xn) + ®zn zn+1 = ¡xn (2) x = ¡13=5 x + x3 + ®z z = ¡x (3) n+1 n n n n+1 n 2 3 2 3 ¡Cx0 + SC z0 + S F1 where the ¯rst and last are analogs of the H¶enonand = 4 0 5 ; (8) 3 2 Du±ng maps, and the middle a two-dimensional version ¡Sx0 + C z0 + CS F1 of the logistic map. For a range of parameter ®, these systems are chaotic with fractal attractors. For ® = 0, where we have used the fact that F2 = ¡x. 3 these maps have one-dimensional attractors in the x; z In order to embed this in R , we displace the function plane because the points of the iteration are of the form in the x direction by an amount (in this case 1) su±cient to include the attractor in the left half plane and to avoid (xn+1; ¡xn), a clockwise rotation of the functional form ¼ self-intersection, and rotate counterclockwise about the of the map (xn; xn+1) by 2 . Thus, the exact functional form of the attracting sets are known, and the projection z axis: of the dynamics onto the x axis give the dynamics of the · ¸ · ¸ original one-dimensional map. h i 2 3 x cos(2¼t) ¡ sin(2¼t) 0 1+Cx0+SC z0+S F1 y = sin(2¼t) cos(2¼t) 0 0 = z 3 2 0 0 1 ¡Sx0+C z0+CS F1 2 3 THE EXPLICIT SUSPENSION 2 3 cos(2¼t)(1 + Cx0 + SC z0 + S F1) 2 3 = 4 sin(2¼t)(1 + Cx0 + SC z0 + S F1) 5 : (9) 3 2 Let xn+1 = F1; zn+1 = F2 = ¡xn be the components ¡Sx0 + C z0 + CS F1 of our two-dimensional mappings of the plane. Then the action of the maps can be broken down into two geomet- To form a di®erential equation we solve (8) for the ini- rically distinct steps. First the (x; z) plane is deformed tial conditions in terms of the variables x and z, take 3 the derivative of (9) with respect to the interpolating pa- rameter t, and substitute the expressions for the initial conditions into the di®erential equation. At this point we have non-autonomous di®erential equations. To make them autonomous, we use the half angle formulas twice to write all interpolation and rota- tion operators in a form so that their period is 2¼t for t 2 [0; 1]. Thus, let ( q p 1 + 1 2+2 cos(2¼t) 0·t· 1 C = q 2 4 p 2 (10) 1 ¡ 1 2+2 cos(2¼t) 1 ·t·1 2 4 2 FIG.