Physica A 191 (1992) 564-570 North-Holland

Fractal structures of spheroidal chaotic attractors

Michael Klein” and Gerold Baierb “Institute for Physical and Theoretical Chemistry, University of Tiibingen, Morgenstelle 8, W-7400 Tiibingen, Germany

blnstitute for Chemical Plant Physiology, University of Tiibingen, Corrensstrafle 41, W-7400 Tiibingen, Germany

A three-dimensional generic map exhibiting spheroidal attractors of all types of dynamics possible in three dimensions is introduced. The map is designed to easily invert the stability features of the spheroidal attractors giving rise to repellers or basin boundaries of locally similar geometric properties. For the three different types of ordinary chaos with one positive Lyapunov characteristic exponent a criterion is provided for the close correlation between chaotic dynamics and fractal structures.

1. The system

We analyse the following 3-variable generic map [l]:

xi+l =uxiYi +.ff(‘i7 b)) Yj+l =xi 3 zi+l =Yi > (1)

where x, y, z, a, b E IF!; i E N and a >O, b 2 1. The dissipation function

f(z, b) = -bz + (b - 1)z3 (2)

is controlled by the dissipation parameter b. The quadratic nonlinearity in the first variable x is governed by the control parameter a. The second and third variables y and z are simple linear delays.

The conservative, volume-preserving case of eq. (1) with f(z, b = 1) = -z illustrates the generating mechanism of the dynamics. The origin is a center with one real eigenvalue and a pair of complex-conjugated eigenvalues follow- ing from (AZ, = -1). All bounded solutions of the system live on spheroidal but non-toroidal hyperplanes around the origin. Depending on the initial conditions (x,, yO, z,) and by variation of the control parameter a all types of conservative dynamics may be found. Fig. la shows a conservative chaotic orbit with a spectrum of Lyapunov characteristic exponents (LCEs) of

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M. Klein, G. Baier I Fractal structures of spheroidal chaotic attractors 565

3) b)

Fig. 1. (a) One conservative chaotic orbit of eq. (1) on a spheroidal hyperplane. The quadrant

with x CO, z >O is cut away to visualize the spheroidal structure. The three ellipses are

cross-sections of the attractor at 1x1 CO.001, IyI <O.OOl and IzI <O.OOl. Initial conditions x0 = 1.3,

y, = 0.1, z,, = 0.1; parameters a = 1.25, b = 1.0; axes -1.4. 1.4; orbit with lo6 iterations. (b)

The dissipation function f(z, b) = - bz + (b - 1)z3 of eq. (2) for values of the dissipation parame-

ter b = 1.3 (solid line) and b’ = 0.25 (dotted line). The stability of the points z = *l is inverted due

to the local slope.

(+,O, -), IAll = IA& Th e orbit fills a two-dimensional hyperplane which resembles a tetraoidal surface.

For the dissipative case of eq. (1) with b > 1 the dissipation function f(z, b)

gives rise to three unstable fixed points at (P = ysS = 2”) with

1 Xl ==o, x;s3 = 2(b _ 1) [-a k j/a’ + 4(b2 - l)] . (3)

The origin remains an unstable fixed point with one real and a pair of complex-conjugated eigenvalues following from (AZ, = -b). It is a repellor for b > 1. The chaos-generating mechanism described for the conservative case still creates a spheroidal dynamics. The cubic dissipation function f(z, b) in con- nection with the simple linear delays in y and z guarantee the existence of stable bounded solutions around the unstable origin. The stability conditions of these attractors depend on the local slope of the function f(z, b) at the two

points f(z, b) = --z (see fig. lb).

2. Transition of simple chaotic attractors

The system of eq. (1) exhibits all types of dynamics possible in three dimensions. Especially the first steps on the chaotic hierarchy [2] from ordinary chaos with one positive LCE to hyperchaos with two positive LCEs and hyper2-chaos with three positive LCEs may easily be found.

566 M. Klein, G. Baier I Fractal structures of spheroidal chaotic attractors

Fig. 2. LCE scenario for an interval of the dissipation parameter b with a constant parameter

a = 1.25. The dotted lines are the three Lyapunov characteristic exponents A, (top), A, (middle)

and A, (bottom). The other two lines clearly show the zero-line crossings of the two-by-two sums of

the LCEs (A, + A,) (top) and (A, + A,) (bottom).

Here we focus on the three different ordinary chaotic attractors with only one positive LCE (A, > 0, A,, A, < 0) found with the map (1). The transitions between these three types are found in parameter space where the two-by-two sums (A, + h2) or (A1 + A3) of the ordered LCEs (A, > A, B AX) change sign

(fig. 2). The different fractal structures of the ordinary chaotic dynamics may be

explained by a competition between the local exponential divergence (the source of chaos) and the local exponential convergence (attraction). Diver- gence is tangential to the unstable manifolds of the fixed points with a local positive exponent of absolute value 1 A:‘(. Local convergence may be found along the two directions orthogonal to the unstable manifold. We name the absolute value of the smaller of the two negative local exponents ) A:‘] and the absolute value of the larger negative local exponent 1 A:‘).

With b = 1.3, a = 1.25 there is simple chaos with A, > 0 and (A, + A,) < 0, which implies (A, + AX) < 0. The numerically calculated dimensions (see table I) confirm the impression of the cross-section (fig. 3a). The simple chaotic attractors have a striated fractal structure generated by one smooth, infinitely often folded line. The fractal dimension is less than two. With (A1 + AZ) < 0 the local contraction along ]A:“‘] and ]A:“‘/ 1 a most everywhere exceeds the local divergence with 1 A:‘]. Therefore the state space is locally contracted to lines parallel to the unstable manifolds.

With b = 1.2, a = 1.25 we find the so-called Kaplan-Yorke chaos. The computed dimensions (table I) and the close-up of the cross-section (Fig. 3b) confirm the Kaplan-Yorke conjecture [3], which means that the fractal dimen- sion of the attractor is increased above the nearest integer value. The chaotic attractor beyond the Kaplan-Yorke transition ((A, + A*) > 0, (A, + A3) < 0),

M. Klein, G. Baier / Fractal structures of spheroidal chaotic attractors 561

Table I

Numerically calculated Lyapunov characteristic exponents (LCEs) and dimensions for the three

different ordinary chaotic attractors of eq. (1). D, Lyapunov dimension gained from the LCE

(after Kaplan and Yorke [3]), correlation dimension D,,, (after Grassberger [4]), information

dimension D,,, and capacity dimension DEap (after Liebert [5]). All values gained from 30000

points.

Parameter LCE D, D COT D I”f D cap

a = 1.25 A, = +0.152 1.86 1.68 1.81 1.91

b = 1.3 A, = -0.176 20.01 i-o.01 50.02 20.01 A, = -0.509

2 0.001

a = 1.25

b=1.2

A, = +0.157 2.10 2.07 2.12 2.16

A, = -0.134 kO.01 kO.01 kO.05 -co.02 A, = -0.217

? 0.001

a = 1.25 A, = +0.158 2.41 2.51 2.37 2.42

b = 1.1 A, = -0.092 -co.01 20.05 kO.01 20.02

A, = -0.140

r0.001

may be described as a smooth, infinitely often folded sheet (“folded curtain”). In this case the local divergence with \A:“‘1 on the average dominates the local convergence with 1 A:‘[. In the vicinity of the unstable manifold the orbits are no longer contracted to a line but may locally diverge into the plane orthogonal to the second negative local exponent A!“. The Cantor-set line-segments of the cross-section (fig. 3b) reveal the sheet-like structure of the attractor now having fractal dimensions between two and three.

I)

56 ..I

kd

I.56 1 0.98 1.12 ] L

1 j-0.56 0.98

_(I

0

3

36

Fig. 3. (a) Close-up of the (n, z)-cross-section with (1 y] S 0.001) of the simple chaotic attractor

revealing a pointwise Cantor-set structure (a = 1.25, b = 1.3). (b) Close-up of the (x, z)-cross-

section of a Kaplan-Yorke-type chaotic attractor (a = 1.25, b = 1.2) with a Cantor-set structure

made of line-fragments.

568 M. Klein, G. Baier I Fractal structures of spheroidal chaotic attractors

4 . .

1.56 (x,4

-0.56 0.98 pw--+- c ._ 1.26

(b)

Fig. 4. (a) Close-up of the (x, z)-cross-section of the “bi-fractal” chaotic attractor with a = 1.25, b = 1.1, revealing a complicated nowhere differentiable microstructure. (b) Two-dimensional

close-up of the spheroidal basin boundary of eq. (1) with f(z, b’), b’ = 0.875. Exploding points are

white, the basin of attraction of the finite attractor is black. Scan in the (n,, z,)-plane of initial

values (y, = 0). Note the close resemblance with the same area of the attractor shown in (a).

With b = 1.1, a = 1.25 a new feature for the attractors appears. Besides (A, + AZ) > 0 now the sum of the first and the third LCE, (A, + h3) > 0, is positive, too. With the second transition the chaotic orbit becomes more space filling, the fractal dimensions of the attractor are between two and three (table I). But, more interesting, the spheroidal attractors are fractalized in a nowhere differentiable manner (fig. 4a). There is an aditional fractalization of the attractor along the direction of the second negative local exponent A:‘. The negative local exponents 1 AZ’1 and 1 A:‘] are both nearly everywhere smaller in absolute value than the local divergence with 1 A:‘]. The folded sheets are additionally fractalized to yield a nowhere differentiable attractive set. This set may be characterized as “bi-fractal” according to the classification of Rohricht et al. [6]. This second transition also concides with the “phase transition” of hyperbolic dynamical systems described by Paoli et al. [7].

3. Basin boundaries from inverted attractors

Obviously it is rather difficult to establish neighborhood relations for fractal structures generated from chaotic orbits [8]. To find further support for our conjecture of the existence of nowhere differentiable “bi-fractals” we use a special property of the design of our spheroidal map. The chaos-generating mechanism of the map (1) (compare the case of the conservative dynamics with b = 1) and the dissipation function f(z, b) (eq. (2)), which determines the structure of the orbits, are mainly independent by design. The stability

M. Klein, G. Baier I Fractal structures of spheroidal chaotic attractors 569

conditions of the dissipation function f(z, b) at the two points (z = 21) may easily be inverted. We substitute

where the parameter b’ is achieved from the condition of the slope at the fixed points z = +l of the dissipation function. A spheroidal repellor now replaces the former attractor, This basin boundary separates the attractor at the origin and at m (compare fig. lb). Due to the condition of eq. (4) the local properties on the spheroidal attractors are equivalent to the situation on the basin boundary though the stability is inverted. Fig. (4b) shows a close-up of the basin boundary gained as two-dimensional scan in the three-dimensional space of initial values (x0, y,, zO) with f(z, 6’).

4. Summary

PI.

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570 M. Klein, G. Baier I Fractal structures of spheroidal chaotic attractors

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