Non-strange chaotic attractors equivalent to their templates

John StarrettDepartment of Mathematics

New Mexico Institute of Mining and Technology801 Leroy Place, Socorro NM, 87801 USA

We construct systems of three autonomous first order differential equations with bounded two-dimensional attracting sets M . The flows on M are chaotic and have one dimensional Poincaresections whose Poincare maps are diffeomorphic to maps of the interval. The attractors are twodimensional rather than fractal, and when “unzipped” they are topologically equivalent to thetemplates of suspended horseshoes.

INTRODUCTION

The analysis of the dynamics of chaotic systems can bebroken roughly into two parts – the quantitative, whichemphasizes measures such as Lyapunov exponents, frac-tal dimensions and power spectra [1], and the topological,where qualitative characterizations such as templates andbounding tori are considered [2]. When we think of theinfinite 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 toBirman and Williams [3] [4], a branched two-manifoldthat results when the attractor is collapsed along its sta-ble manifolds. The template supports a semi-flow whichcontains the same periodic orbits as the original system,with at most two additional extraneous orbits.

Extracting a template from a strange attractor is arelatively straightforward, but inexact process. In thesimplest cases, one can guess the form of the templateby making a visual inspection of the attractor and itsPoincare sections to try to determine how the stretch-ing and folding is taking place, see Fig. (1). Once atemplate structure is proposed, the knots and links thatare supported by that structure can be determined. Onethen attempts to verify that the periodic orbits corre-sponding to these topological knots and links exist in thedynamics of the original system. This verification pro-cess is the source of the inexactness: one never knowsif one has checked a sufficiently large set of orbits. Formore complicated systems, one can extract a set of peri-odic orbits by the method of close returns or by Pyragas[5] or similar chaos control methods [6], and determinea set of topological invariants such as linking numbersand relative rotation rates for these orbits. One can thenconstruct a template based on symbolic dynamics givento the orbits [7].

Because template construction can sometimes be a dif-ficult iterative process, it is an interesting exercise to con-struct an explicit chaotic dynamical system in the formof a system of three autonomous differential equationswhose dynamics actually take place on a template. Ourapproach is to suspend a two-dimensional chaotic dynam-

FIG. 1: On the left, a portion of a chaotic orbit of the Rosslersystem, on the right, a template for the Rossler system withcounterclockwise flow corresponding to counterclockwise flowabout the z axis in the attractor.

ical system whose attractor is one-dimensional. Such anattractor cannot be called “strange” because it does nothave a fractal structure. The two-dimensional maps wesuspend are diffeomorphic to one-dimensional maps. Be-cause the exact Poincare map dynamics are known, suchcontinuous chaotic systems might be useful test beds forunderstanding the behavior of map based and continuouschaos control methods, and other areas where the rela-tion of a Poincare map to its flow might be useful. Also,these attractors are the infinite dissipation limiting caseof families of strange attractors.

SUSPENDING MAPS TO FLOWS

The first detailed exposition of the conditions thatmust hold for an explicit suspension of a map to a flowwere given by Channell [8]. Mayer-Kress and Haken [9]suspended a Henon map to a cylinder ℝ2 × I, but theresulting differential equations could not be integratedforward normally: the integrator must reset t whenevert = 1 and load in the final conditions at t = 1 as thenew initial conditions each time. Recently Nicholas[10]described a method for making an exact autonomoussuspension of a map in an open half plane to a flow.

2

FIG. 2: Left to right, the attractors of the quadratic map,logistic map and cubic map, in black, along with the rotatedfunctions along which the attractors lie, in gray.

The construction embeds a flow on a quotient spaceQ = ℝ2 × [0, 1]/ ∼, where 0 and 1 are identified in[0, 1], in ℝ3 by wrapping it around the z axis. Moreexactly, the flow Á on Q = ℝ2 × S1 is embedded in ℝ3

by truncating ℝ2 to (−a,∞) × ℝ and constructing ℝ3

as (0, 0, z) ∪ ℱ((−a,∞) × ℝ), where ℱ((−a,∞) × ℝ) isa book-leaf foliation of ℝ3/(0, 0, z) rotationally symmet-ric about the z axis. The method in [10] was demon-strated on two-dimensional maps with strange attractors,a Henon and a Duffing map. While this method only sus-pends the dynamics in the right half plane, it produces asystem of differential equations that can be integrated ina normal fashion. We use this method to suspend two-dimensional maps with one-dimensional attracting sets,resulting in attractors with no fractal structure, but withverifiable chaotic dynamics.

Consider the difference equations

xn+1 = 4/3− 5/4x2n + ®zn zn+1 = −xn (1)

xn+1 = 4xn(1− xn) + ®zn zn+1 = −xn (2)

xn+1 = −13/5xn + x3n + ®zn zn+1 = −xn (3)

where the first and last are analogs of the Henon andDuffing maps, and the middle a two-dimensional versionof the logistic map. For a range of parameter ®, thesesystems are chaotic with fractal attractors. For ® = 0,these maps have one-dimensional attractors in the x, zplane because the points of the iteration are of the form(xn+1,−xn), a clockwise rotation of the functional formof the map (xn, xn+1) by ¼

2 . Thus, the exact functionalform of the attracting sets are known, and the projectionof the dynamics onto the x axis give the dynamics of theoriginal one-dimensional map.

THE EXPLICIT SUSPENSION

Let xn+1 = F1, zn+1 = F2 = −xn be the componentsof our two-dimensional mappings of the plane. Then theaction of the maps can be broken down into two geomet-rically distinct steps. First the (x, z) plane is deformed

by

[xn+1

zn+1

]=

[ −F2

F1

], (4)

then the result is rotated by ¼/2 clockwise

[xn+1

zn+1

]=

[cos(¼2 ) sin(¼2 )− sin(¼2 ) cos(¼2 )

] [ −F2

F1

]=

[F1

F2

]. (5)

We suspend the maps to flows by continuous interpola-tions between initial and final conditions. The interpola-tion is performed using the basic interpolating functionsC = cos(¼2 t) and S = sin(¼2 t), which also appear in a

rotation matrix R =

[C S

−S C

]. As interpolating diffeo-

morphisms, these are

C2 = cos2(¼2t)

(6)

S2 = sin2(¼2t)

(7)

taking initial to final conditions by C2x0+S2x1 as t variesfrom 0 to 1. Because of the scaling of the argument, aquarter clockwise rotation and a smooth interpolationbetween initial and final conditions is accomplished as tgoes from 0 to 1, and this suspends the map to a cylinderℝ2 × I.In anticipation of our embedding in ℝ3, we write the

suspension to a cylinder ℝ2 × I as⎡⎢⎢⎢⎢⎢⎣

xyz

⎤⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎣

C 0 S0 1 0−S 0 C

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

C2x0 − S2F2

0C2z0 + S2F1

⎤⎥⎥⎥⎥⎥⎦=

=

⎡⎣−Cx0 + SC2z0 + S3F1

0−Sx0 + C3z0 + CS2F1

⎤⎦ , (8)

where we have used the fact that F2 = −x.In order to embed this in ℝ3, we displace the function

in the x direction by an amount (in this case 1) sufficientto include the attractor in the left half plane and to avoidself-intersection, and rotate counterclockwise about thez axis:

[xyz

]=

[cos(2¼t) − sin(2¼t) 0sin(2¼t) cos(2¼t) 0

0 0 1

] [1+Cx0+SC2z0+S3F1

0−Sx0+C3z0+CS2F1

]=

=

⎡⎣cos(2¼t)(1 + Cx0 + SC2z0 + S3F1)sin(2¼t)(1 + Cx0 + SC2z0 + S3F1)

−Sx0 + C3z0 + CS2F1

⎤⎦ . (9)

To form a differential equation we solve (8) for the ini-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 initialconditions into the differential equation.

At this point we have non-autonomous differentialequations. To make them autonomous, we use the halfangle formulas twice to write all interpolation and rota-tion operators in a form so that their period is 2¼t fort ∈ [0, 1]. Thus, let

C =

{√12+

14

√2+2 cos(2¼t) 0≤t≤ 1

2√12− 1

4

√2+2 cos(2¼t) 1

2≤t≤1(10)

S =

{√12− 1

4

√2+2 cos(2¼t) 0≤t≤ 1

2√12+

14

√2+2 cos(2¼t) 1

2≤t≤1(11)

Then, using the identity cos(2¼t) = x√x2+y2

, and the fact

that 0 ≤ t ≤ 1/2 ⇒ y ≥ 0, we obtain

C =

⎧⎨⎩

√12+

14

√2+ 2x√

x2+y2y≥0

√12− 1

4

√2+ 2x√

x2+y2y<0

(12)

S =

⎧⎨⎩

√12− 1

4

√2+ 2x√

x2+y2y≥0

√12+

14

√2+ 2x√

x2+y2y<0

(13)

Substitution of these into the differential equation resultsin autonomous differential equations that can be numer-ically integrated in a normal fashion: one does not haveto reset t to zero and substitute the final conditions inas new initial conditions to integrate the solution for-ward past t = 1 as in the usual case of a suspension to acylinder[9].

The suspension of these one-dimensional attractingsets results in a two-dimensional attracting set, whichis the unstable manifold of the system. This manifold,though, is a branched two manifold because the stretch-ing and folding of the original map attractor is two toone. The branches join smoothly at the y = 0 plane ina one-dimensional set. At the critical point of the mapthere is a non-differentiable fold. To form a templatefrom this stable manifold, we “unzip” it by flowing thecritical point(s) at the Poincare section backward in timeto the previous Poincare section and removing this pieceof orbit. What remains is a branched two-manifold, atemplate in the usual sense of the word, except for theopen boundaries where the orbit of the critical point(s)were removed. If we close this set, we have a specific pa-rameterized template. In the case of the logistic suspen-sion, the template is (topologically) exactly the templateof the suspended Smale horseshoe.

FIG. 3: On the left, the suspended quadratic map, and onthe right, the corresponding template.

FIG. 4: On the left, the suspended logistic map, and on theright, the corresponding template.

FIG. 5: On the left, the suspended cubic map, and on theright, the corresponding template.

BIFURCATIONS

The period doubling bifurcation sequence of the sus-pension of the logistic map is shown in figure (6). Becausewe know the exact bifurcation sequence of the logisticmap, we know the related bifurcation sequence of its sus-pension. We can use this knowledge to determine theexplicit expressions for the parameterized surfaces com-prising the template through its bifurcation sequence.

4

FIG. 6: From top to bottom, left to right, periods 1, 2, 4, 8,chaos and period 3 .

BASIN OF ATTRACTION

The basins of attraction of these maps are especiallysimple, as are the basins of the suspended attractors.As an example, consider the Poincare section of thelogistic suspension at y = 0. For the original mapxn+1 = 4xn(1− xn), zn+1 = −xn under iteration, pointswith x coordinate in [0, 1] stay in that interval for alltime, while points outside it go to −∞. The basin ofthe map in the z direction is unbounded, as no matterwhat the value of z in the point (x, 0, z), it will have thevalue −x after one iterate. Thus, all points in the verticalstrip x ∈ [0, 1], y = 0, z land on the attractor after onefundamental time unit. For the suspension, however, theinterpolation includes a rotation and is valid only for theright half plane, so there exist points with z < −1 in thebasin of attraction of the map that map into the z axisover the course of a cycle, and therefore are not in thebasin of attraction of the suspended system.

STABLE MANIFOLDS

Because the differential equations for these systems arederived from explicit expressions for the solutions, theformulas for the stable manifolds can be expressed ex-plicitly as parameterized surfaces in parameters s and t,with s parameterizing the z coordinate and t the inter-polation and rotation parameter as usual. Thus, for anyinitial x0 ∈ [0, 1], 0 ≤ t ≤ 1 and appropriate range of s(avoiding z values out of the basin of attraction),

⎡⎢⎢⎢⎣

cos(2¼t)(1 + x0 cos ¼2 t+ s sin ¼

2 t cos2 ¼

2 t+ sin3 ¼2 t F1)

sin(2¼t)(1 + x0 cos ¼2 t+ s sin ¼

2 t cos2 ¼

2 t+ sin3 ¼2 tF1)

−x0 sin ¼2 t+ s cos3 ¼

2 t+ cos ¼2 t sin

2 ¼2 t F1

⎤⎥⎥⎥⎦

gives a portion of the stable manifold that collapses,at t = 1, onto the orbit with initial condition(F1(x0), 0,−x0).

FIG. 7: A cut through the attractor showing the portions ofthe stable manifolds of three orbits of the logistic suspension.

SUMMARY

We have constructed systems of autonomous differ-ential equations for continuous chaotic systems in ℝ3

with non-fractal attractors that are, up to an unzip-ping, topologically equivalent to their template represen-tations. The stable and unstable manifolds can be ex-plicitly given as parameterized surfaces, and the solutioncurves can be explicitly given as parameterized curves.

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