Chaos, Sditions & Fracra/s, Vol 9, k-a 3. pp. 437--14X. IYYti 0 1998 Elsevier Science Ltd. Ail rights reserved

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PII: SO960-0779(97)00113-6

Groups and Nonlinear Dynamical Systems: Chaotic dynamics on the W(2) x W(2) group

KRZYSZTOF KOWALSKI and JAKUB REMBIELIkXI

Department of Theoretical Physics, University of todi, ul. Pomorska 149/153,90-236 todi, Poland

(Accepted 25 April 1997)

Abstract-In our previous paper, Groups and nonlinear dynamical systems, dynamics on the SU(2) group, in Physica D, 1996, 99, 237, we introduced an abstract Newton-like equation on a general Lie algebra such that submanifolds fixed by the second-order Casimir operator are attracting sets. The corresponding group parameters satisfy the nonlinear dynamical system having an attractor coinciding with the submanifold. In this work, we discuss the case with the Su(2) x SU(2) group. The resulting second-order system in R6 is demonstrated to exhibit chaotic behaviour. 0 1998 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

One of the most important problems in the theory of dynamical systems relates to finding and classifying attracting limit sets or simply attractors. Another important problem is how to study the system on its invariant manifold. In our previous work [l], we introduced a method for the construction of G-invariant nonlinear dynamical systems having an attracting set coinciding with a submanifold generated by the second-order Casimir operator referring to a given Lie group G. The example of the S&‘(Z) group discussed therein led to the oscillatory dynamics on the orbit, i.e. the sphere &. As suggested in [l], such regular dynamics need not be the case for groups with higher-dimensional submanifolds. In this paper, we examine the case with the SU(2) X SU(2) group. Namely, following the general scheme described in [l], we introduce the nonlinear second-order system in R6 satisfied by the group parameters. We then show that the system exhibits chaotic behaviour on the sphere S5 which is a submanifold fixed by the second-order Casimir operator corresponding to the SU(2) X SU(2) group.

2. THE NEWTON-LIKE EQUATION

In this section, we recall the abstract Newton-like equation on a general Lie algebra such that submanifolds fixed by the second-order Casimir operator are attracting sets [l]. This equation generates the nonlinear dynamical system satisfied by the group parameters, having an attractor coinciding with the submanifold. Consider the following second-order differential equation on a Lie algebra g:

pX + v;i: + pX + aY = eixYe-ix, X(0) = X0, X(0) = X,, (1) where X(t): R+ IJ is a curve in g, Y E Q is a fixed element, p.,v,p,r E R, and dot designates differentiation with respect to time.

Let us assume that p # 0. Clearly, without loss of generality we can set /.,L > 0. On demanding that eqn (1) admits the solution on the submanifold fixed by the second-order

437

Casimir operator of the form tr x ’ -. constant -p 0.

resealing t ---) VGr and setting r/i \/-i -- p. we arrive at the Newton-like equation

lr X’ .U + /3 i 3. 3 X - e’.‘Yc Ix - Y. X(0) -= X,,. .%((I) = k,,.

(Using eqn (3). we find that

That is, if fi > 0 then the solution to eqn (I ) approaches the s&manifold

Jr can be easily checked that, for p 5 0. there is no solution on the submanifold. The only exception arc the initial data such that

t r .Y,, x,; = 0. (6) A more detailed analysis of eqns (3) and (4) is provided in [I]. Notice that in the general

case of an n-dimensional compact Lie algebra the manifold defined by eqn (2) is simply the sphere S,, :.

3. DYNAMICS ON THE W(2) x W(2) GROUP

Our aim now is to discuss the nonlinear dynamical system implied by the abstract Newton- like equation in the case of the SL’(2) x SC(Z) group. We first observe that due to the product structure of the group considered. the general elements of the Lie algebra X(r) and Y can be written in the form

where X(r)-,%‘.(/)‘.Y (t).Y -)‘. ’ Y . (7)

[X.X ]=O.[X .Y /:-z-(X .Y.)--0.Ir.Y X -ztrX.X -0. (8) Notice that in view of eqns (7) and (8) the manifolds given in eqn (2) take the form

rr(X’ t X’ ) -z constant. (‘1) We now return to eqn (2). From eqns (3). (71 :tnd (8). wc get

.I’, (0) ,% (J_ A’ (0) = ,%’ ,,. .u (0) - .i. ,). ji (0) -- ,* (,. (IO) Now consider the following realization of the generators of the SU(2) X SU(2) group:

(11)

Groups and nonlinear dynamical systems 439

where u = ((T~,u~,(T~) and (TV, for i = 1,2,3, are the Pauli matrices. Clearly, we can write the general elements of the Lie algebra X,(t) and Y, of the Lie algebra as

X+(t) = x+(t).J+,X-(t) = x-(Q-J-, Y, = a+.J+, Y- = a-.J-, (12) where x,(t): R+ R3, a, is a constant vector in R3 and the dot denotes the inner product. Substituting eqn (12) into eqn (lo), we obtain the nonlinear system of second-order equations

ii, + pk+ + i:+2 x”, + x2

x+ = (cos 1x+( - l)a+ + * a+ Xx+

(a+*x+)x+ +(1 --oslx+I) x”, 9

f- + /3%- + ir:+jY!. x2 + x2_ x- = (cos Ix-1 - l)a- + - sin Ix-l a_ x x

+ IX-I

+ (1 - cos 1x-l) (a-.x-)x-

X2 ’

x+(O) = X+0, x-(O) = x-0, k+(O) = k,,, k-(O) = s-0, (13) where X denotes the vector product, and 1x1 = V’? is the norm of the vector x.

Notice that the manifolds given by eqns (9) and (12) are the five-dimensional spheres

x: + x2 = constant. (14) We now examine the asymptotic behaviour of the system (13). First we observe that (13)

implies that

d -•s++ d . . -*x =-(jp-2

x+.%+ + x-*ir- a+ dz;r- - x”, + x2 .

Thus as the solution to (13) when p > 0 approaches the sphere given by eqn (14) the system (13) becomes dissipative one with exponential contraction of a volume element. Furthermore, it can be easily checked that the SU(2) X SU(2) realization of eqn (4) takes the form

x: + x2 = 2(x+“‘jY+o + ~-,k,) $- (1 - epPt) + x:, + xTo.

Thus whenever the initial data satisfy the inequality

$ (X+o~~+o + x-&o) + x”+n + xto>o,

and p > 0, then the solution to (13) approaches the sphere S5 given by

2 x: + x2 = - (x+&+(J + X-“&J + x:0 + X2@

B

Evidently, the sphere S5 is an invariant set for the initial conditions such that

X+()‘jY+() + x-&o = 0

(17)

(18)

(19)

44 I K. KOWALSKI and .I. KF.MRIELINSKI

and arbitrary p. ‘The solutions corresponding to the remaining initial data do not approach the sphere S,. Indcctl. if p -> 0 and

;(x.,,.k+,; - x ,,‘i ,,) + x.i,, + XT ,, -- 0. X.,)‘X >(, + x ,,-k (, z 0. (20)

then the solutions to (‘ITi tend asymptoticall!; to the singular point x. =O, x =O. Furthermore. if p -> 0 and

;(x .,,. %.,,-1, x ,,.i. ,,) -t X’.,, A- x- ,, c.- 0. (21)

then the singular point is approached after a finite period of time

Finally, for fi 5 0 and x , ,,-x , (I -t x ,,-X ,, it 0. the trajectories go to infinity. We now return IO (13). Notice that in view of the covariance of eqn (IO) with respect to

the group transformations. we can. without loss of generality. in (3.6) set

a :: (0.0.~ ~‘0s CY ). a = (0.0.~ sin tr). (‘3)

Thus we finally arrive at the system

x , (0) z- x , (,. k ! (0) -= i , (,. (24)

where ~1,~ = u cos (Y and a. .1 =R sin a. Notice that the parameter (Y can be restricted to the interval [O,7r/2]. Indeed. taking into account eqn (2.1). wc find that the transformation it -.A (I + ~2 leads to

(/ . : --* (I ,. (1 -- I/ (. (25)

or. in view of cqn (24).

Y . ..y .\’ ..* .\ I. 1. , .-.a \ 1. .v. : --+ .\ :. (26)

Further. we observe that for u -: 7i:4 the system (24) is symmetric: in x. and x- variables. l-‘or an easy illustration of this observation. see Fig. 2. It should also be noted that. by virtue of the transformation law ot‘eqn (3). under the scaling t-+ AI.

\\c have actually two bifurcation parameters /?.‘irr. whcrc o - 0. and tl. In order to &cuss the symmetries of eqn (24). consider the original abstract eqn (j).

Evidently. rqn (3) is invariant under the transformations referring to the stability group of 1’. that is. transformations leaving Y unchanged. Hence. making use of eqns (IO) and (12). we

Groups and nonlinear dynamical systems 441

(a) 1

0

-1

* G-2

-3

-4

-5

.:, ;: ‘. ,, .,, . . . .

.,,- :

‘. .::.

-3 -2 -1 0 1 2 3

4

3

2

1

-0 k

-1

-2

-3

-4

-5

(b)

I -4 -3 -2 -1 0 1 2 3 4

X-1

Fig. 1. The system (24) with /3=0.1, a=OS, (~=0.2, x+,=(1,1,1.5), x-,=(1,1,1.5), i+0=( - O.l,O.l,O.l). k-,, = (O.l,O.l,O.l). (a) The projection of the Poincare section, of the attractor on the (x+,,x+,) plane. This section, and the following ones in the case with the projection on the (x+,x+~) plane, are defined by the hyperplane n(x - x@)) = 0, where n is the normal vector and x(O) IS a point of the hyperplane, n,x@‘)& E R’*, n = (O,l,O,...,O) and x(“)= 0. (b) The projection of the Poincare section of the attractor on the (x-,,.x-~) plane. This section, and the following ones in the case with the projection on the (x-,,.-J plane, are defined by the hyperplane n(x - x(O)) = 0, where n = (O,O,O,O,l,O ,..., 0) and x @) - 0 The Lyapunov exponents are A, = 0.02, AZ = A3 = A, = A5 = f 0.00, A, = - 0.01, -

~,=-0.06,/\~=-0.08,A~=-0.09,A,~=-0.10,A~,=-0.12,A~~=-0.16.~eLyapunovdimensionis6.1.

tind that solutions to eqn (24) have the form invariant under rotations about the .r* 1 and s : axes such that

This suggests that, whenever the initial condition x.,,, x ,), x,,, and ir ,, corresponds to an attractor A. the basin of attraction of A contains the points x’. iI. x’ ,,. x’, ,, and ir! ,, of the form

.Y I, ,,, = cos p1 .t- : !,, + sin cc: = .y 2 .w’ -r ‘, ?,, = .- sin cp ?.t ! :,, + cos 9 , ri ‘:,, s ‘C :,, = .Y, ,,). (2’))

\ ‘, (I, == cos qzt i ! IO t sin cp , .*+ :,),.i- ‘f ?,I = - sin cp , .i I ,,, -t cos p 1 ,i , ,,,.i ‘f :,) = .C I ?I.

cF.4 E lW71). (30)

Thus the basin of attraction contains two circles given by eqn (29) such that ,) ‘2 > , x;,, ;; x .,,, x’.;, y; X‘ j,. (31)

and two vector fields of eqn (30) obtained from k,” by rotating this about the s-.; and s 1 axes. by the angle referring to the position of the corresponding point of the circle.

In Figs 1 and 2. we show examples of strange attractors from numerical integration of the cvstem (24). As expected these attractors are svmmetric under rotations about the s,: and .I : axe>.

It follows from the computer simulations illustrated in Fig. 3 that in the parameter space of the system (24) in a neighhourhood of the attractor from Fig. 1 thrrc exists a quasiperiodic trajectory.

A look at Fig. 3 is enough to conclude that m the case with the attractor from Fig. 1 we deal with the quasiperiodicity to chaos transition as in the Ruelle-Takens-Newhousc scenario 12. 31. In th e cirsc of the attractor from Fig. 2. WC‘ have most probably the new scenario.

Namely. it turns out that there exist in the parameter space of the system (24) two nearby quasiperiodic orbits shown in Fig. 4. The computer simulations suggest that these quasiperiodic orbits arc separated by an infinitesimal perturbation of the bifurcation parameter ,G. As a consequence of the infinitesimal nature of the perturbation. the irregular transitions occur between the chaotic attractors arising from the perturbation of the quasiperiodic orbits from Fig. 4(a) and (b). respectively. These attractors arc shown separately in Fig. 5. We remark that the attractors have the same form as those shown in Fig. 6 arising in the transient chaos before reaching the quasiperiodic state. The authors did not find such a scenario of transitions between chaotic attractors in the literature. As the bifurcation parameter decreases and approaches the value corresponding to the attractor from Fig. 2. the frequency of transitions between the two attractors increases. On the other hand. the decay of the bifurcation parameter leads to the deformation of the attractors. As a result of these two combined processes, we arrive at the attractor illustrated in Fig. 2. In this sense. that attractor arises from the perturbation of two quasiperiodic states.

4. CON<‘I.USION

In the present work H’C’ have studied the particular SU(2) x Sc’(2) realization of the abstract Newton-like equation having attractors coinciding with the submanifolds fixed by the second-order Casimir operator referring to a given Lie group. The resulting second-order system in iI@’ has been shown to exhibit chaotic behaviour. The concrete examples of the chaotic attractors with the SU(2) x SU(2) ‘y s mmetry have been provided. One of them (see Fig. 2) is most probably related to the new scenario of transitions between chaotic attractors

Groups and nonlinear dynamical systems 441

6

4

2

0

-2 4 2

-4

-6

-8

-10

-12

6

4

-6

.‘.

(a) _ .,

‘; :

..’ __:: . .

‘: “‘, . . ,’

(b) ‘, :

Fig. 2. The system (24) with p=O.l, a=l, a=rr/4, x+,=(2,2,2), x-,=(2,2,2), i+o=(O.l,O.l,- 1). t-,=(1,1,1). (a) The projection of the Poincare section of the attractor on the (x+~,.x+~) plane. (b) The projection of the Poincare section of the attractor on the (~-~,.-a) plane. The Lyapunov exponents are A, =0X%, ha=O.O1, A3=A4=As=A6=f0.00, A,=-0.04, As=-0.07, As=-0.10, A,,=-0.11, All=-0.14, A,?=-0.21. The Lyapunov

dimension is 7.4.

K. KOWhtSKl and J. KEMBIFLl

Fig. 3. The system (24) with ,8 = 0. I 17. I’hc remainrng paramelers and thq initial condition are the same as. in Fig. 1. (niThe projection of the Poincare section ofthe quasiperiodic attractor on the (.r r~i. .L - 3 ) plane. (b) The projection

of the Poincar6 section of the quasiperiodic attractor on rhe (.u !. .v :) plane.

Groups and nonlinear dynamical systems 445

0

-2

-8

-10

-1

-2 “+ H

-3

-6

(4

-10 -5 0 5 10

X+1

(b)

-8 -6 -4 -2 0 2 4 6 8

x+1

Fig. 4. The system (24) with (a) p = 0.1102766, (b) /? = 0.1102767. The remaining parameters and the initial condition are the same as in Fig. 2. The projection of the PoincarC section of the quasiperiodic attractors on the (x ~ I. I +D )

plane.

K. KOWALSKI and .I. REMBIELItiSKI

0

-y---------r

-4 !-

t-lg. 5. The transient chaos m the system (ZJ,I with (a) /-i -D.ilU2766, ib) fi 1-0.1102767, before reaching the clua+criodic states from Fig. 3. I’he remaining parameters and the initial condition are the same as in Fig. 2 The

poj~:ctir~~rt of the: Poincnre section ~ri the attractor\ on the (.t ,.I ~‘1 plane.

Groups and nonlinear dynamical systems 447

-4

“+ H

-6

(a)

(b) I l-

O-

-1 -

-2 - 9 H .

-3 -

-4 -

-5 -

-6 -

,,). ‘.. . .

..” .:

.i. ,_‘._ /

‘.‘.

I

-8 -6 -4 -2 0 2 4 6 8

=+1

Fig. 6. The system (24) with p = 0.1099. The remaining parameters and the initial condition are the same as in Fig. 2. For such data, the irregular transitions appear between the attractor from (a) and the attractor from (b). In both

figures, the projection is shown of the Poincare section of the attractor on the (x+~,x+,) plane.

iJh K KOWALSKI and J. REMHIELINSKI

cffectivcly forming one by perturbation of two quasiperiodic states. We point out that the case with the SU(2) x SU(2j realization of the abstract Newton-like equation discussed herein seems to bc the simplest leading to chaotic dynamics. As far as we are aware the investigated system (24) is the first example of the chaolic dynamical system on the manifold determined by the continuous Lie-group structure. We remark that the importance of such an example was indicated in Refs [4-61. Furthermore, the knowledge of chaotic attractors with fixed symmetry and known invariant measure. like those introduced in this work, would be useful for testing methods of detecting symmetry of chaotic attractors, such as for example the method of detectives [7]. Whenever the theory of groups appears to lead to some insight Into the nature of chaos, it seems that the observations introduced herein would be an important point of departure in solving numerous problems.

REFERENCES

i. Kowalski. K. and Rembieli6ski. J.. I’l~~xca II, IYYh. 99, 237. 2. Newhouse. S.. Ruelle. D. and Takens. E. &mm. Ma/h. Phys. 1078. 64. 35. 3. RuelIe. D. and Takens. I:.. C‘on~m. Marh. Phyhys., 1971. 20. 167. 4. Swift. J. W. and Rarany E.. Ibopan J. Mech. B F/uids, 1991. 10. 90 5. Guckenheimtx. J. and Worfolk, I’.. :VoAj,,enCly, 1Y92. 5. 121 1 0. Ekld. M. and Swift. .I. W., :Vc&ineuriry, lYY4. 7, 38.5. T Barany. I!.. l)cllnit/. M. and Goluhitskv. .M.. Pl~vticn I? 190;. 67. hh