Physics Letters A 165 (1992) 231-234 North-Holland PHYSICS LETTERS A

Multifractals in polynomial circle maps

K.M. V a l s a m m a , G. A m b i k a a nd K. B a b u J o s e p h Department of Physics, Cochin University of Science and Technology, Kochi 682022, India

Received 19 September 199 l; revised manuscript received 4 March 1992; accepted for publication 16 March 1992 Communicated by D.D. Holm

The singularity spectrum f(a) and the generaliscd dimensions De of the chaotic attractor associated with the quasiperiodic route in polynomial circle maps are analysed using the perturbative scheme. It is found that Dq in general depends on the order of the inflection point, z, of the map, but Do seems to be independent of z and is equal to I. The f-a curves for large z tend to be relatively flat near the maximum. We also establish a universal relation connecting the minimum and maximum values ofa.

Nonlinear systems with quasiperiodic behaviour, such as coupled oscillators or ring accelerators, can be studied by considering invariant curves in phase space with irrational winding number. In the pres- ence o f strong dissipation the mot ion is confined to the neighbourhood of an invariant circle and the dy- namics can be well approximated by a circle map. As such, circle maps have been subjected to serious and systematic studies in recent years [ 1-5 ]. These maps are unique in that they exhibit the period doubling route to chaos in the mode locked regions or steps and the quasiperiodic ( Q P ) route in between the steps [ 6 ]. The application o f renormalisation group ( R G ) techniques has revealed that universal scaling properties are associated with the quasiperiodic route also and the universality class is determined by the nature of the inflection point at the critical value [ 7 - 9].

Recently a generalised polynomial form has been suggested for the circle map as [ 10 ]

K O,+l=12+O~- ~-~n (2nO,,-2z- 'OlOlZ-l) , (1)

with 0e [ - ½, ½ ]. For z = 3, this map has a cubic in- flection point and therefore belongs to the same uni- versality class as the sine circle map. For ( 1 ), details regarding the variation o f fractal dimension with z have been worked out numerically by Delbourgo and Kenny [ 11 ] for the mode locked regions. The uni- versal indices ot and ~ associated with the quasiper-

iodic route have been recently analysed by Hu et al. [10] .

In this short paper, we try to extend these results by studying the universal behavior as well as the multifractal nature o f the attractors in the quasiper- iodic route, as the order o f inflection z varies. In our study, we try to follow an analytic procedure, in which a perturbative scheme is used to solve the renor- malisation group equations [ 12,13 ]. This has been developed to study the multifractal nature o f the Feigenbaum attractor in one-hump maps [ 14 ] and recently modified to determine aqv and ~ for the sine circle map [ 15 ]. Here we compute relevant gener- alised dimensions as well as f - a values for a set o f z values and attempt to study the nature o f the at- tractor in the limit z-~oo. However, we confine our attention to the golden mean winding number w* and the critical point K = 1.

The RG equations in this context are [ 9 ]

g(x ) = aqpg(g(ol~2x) ) , (2)

g (x ) = a2pg( Otqflg( ot ffpl x) ) , ( 3 )

with g ( 0 ) = 1. Here g(x ) is the universal function and aq~ is the scaling index associated with the se- quence o f rational approximations to the golden mean winding number. Using our perturbation scheme, the expression for g(x ) is obtained as

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Volume 165, number 3 PHYSICS LETTERS A 18 May 1992

1 z - l ~ z g(x )=l+a2~ ' ZO~qp zZa2p] x

z - 1 t .~2z~c2 z (4) 2Z Ogqp

while the scaling factor aqp is obtained from the z equation Dq

i _ l=ot2~ + 2z2---~ p + . . . . (5) OLqp

These expressions are generalisations of those given in ref. [15] for z = 3 . In (5), the asymptotic series in parentheses is replaced by the corresponding Pad6 approximants [L/M]. In an actual calculation, the [L/M] are determined to different orders and the converging value of Otqp is used in further calculations.

The first few iterates o f g ( x ) , starting from Xo= 0, as given by (2) , are

x t = l , x 2 = l / O ~ q p , x3=g(1/Olqp),

x4 = ( 1/O~qp)g(aqp). (6)

These iterates are seen to follow the construction of a Cantor set. At each stage, the probabilities Pi are equal while the lengths li are different. The xi are re- scaled and rearranged to fall in the interval (0, 1 ) as

Xl =1 , X 2 = 0 , X3---- g(l/Ol.qp)--l/OLqp , 1 - ~qp

( 1/tXqp)g(OLqp ) - - 1/Otqp X 4 ---~ ( 7 )

1 -OLqp

I f we now consider the first stage of construction of the Cantor set, the lengths Ii and 12 are given by

ll=xl--X3, /2----X4--X2 , ( 8 )

and the corresponding scale factors are S~ = 1/ll and $2 = 1/12. We attribute equal probability measures to both scales so that P1 =P2 = ½. The equation for r (q ) is then [ 16 ]

S I q-S~ = 2 q . (9)

For any specific value of q chosen, (9) is solved by a root finding procedure to get z(q). Then

Or 2qln2 ce= O q - S~InS1 +SglnS2 " (10)

f

d ... C ..

b a

--(0 -20 0 20 40 q~

Fig. 1. Dqversus q for (a) z=l.1, (b) z=2, (c) z=3, (d) z=4, (e) z=5, (f) z=6.

f(ot) is computed using the relation

f=otq--z . (11)

The generalised dimensions Dq are then computed using

r = (q-- 1 )D, . (12)

Using the computed values obtained as described above, the Dq v e r s u s q curves for a few chosen z val- ues are drawn in fig. 1. It is clear that as z increases, the change from - q to + q becomes more sharp. The Dq values then to converge on the + q side while they are far apart on the - q side. This is reflected in the f - a curve shown in fig. 2. For different values of z, the f - a curves lie close together near the small a re- gion while they are far apart for large a values. Moreover the value of Do, given by the maximum of the f-or curve, seems to be independent of z and is equal to 1. For large z values, say 50 and 100, shown separately in fig. 2b, the f - a curves are compara- tively flatter in the top portion with the exponents crowding along the fiat region. We are led to the con- clusion that as z increases, the points on the attractor show a tendency towards a more or less uniform dis- tribution with the scaling exponents avoiding the most rarefied and concentrated regions. This is in contrast to the behaviour of the Feigenbaum attrac- tor in one-dimensional maps, where as our earlier calculations show [ 14 ], with increase in z, the ex- ponents crowd near the ends, revealing an increasing tendency for bunching.

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Volume 165, number 3 PHYSICS LETTERS A 18 May 1992

1.0

O.8

O.6

f(,z) 0.4

0.2

0.0

tZ > 4 5

(a)

,.o (b)

0.8

0.0

fC=)o., 0.2

0.0 0 El 12 18 24 O0 ~0 42

(Z >

Fig. 2. (a) f-a curves for a few typical z values: (a) z = 1.1, ( b ) z=2, (c) z=3, (d) z=4, (e) z=5, (f) z=6. (b) f -a curves for (a) z=50 , (b) z = 100.

The end points of the f-or curves give the Doo val- ues (ami . ) and the D _ ~ values (amax). In the con- text of quasiperiodic route to chaos, these are given

by [171

In w* In w* D o o = - - D _ ~ = (13)

In a q-pZ, In a qpl "

Using the computed OLqp values, we calculate D~ and D_co from (13). The results are found to agree with those taken from the graph. The relevant results of our computations are collected in table 1. As has been established earlier, our calculations also show that as z - - ,~ , aqp- - , - 1. For large z values, say 50 and 100, the calculated D~ and D_o~ are used to complete the f - a curves in fig. 2b. The variations of Doo and D_o~ with z are shown in figs. 3a and 3b. Doo decreases with z, while D_oo is a monotonical ly increasing function of z. It is clear from (13) that

O / m a x = ZO~mi n . (14)

Our graphical values also establish this relation (ta- ble 1 ). This is a universal relation that is true for any z value and an extremely useful relation since from an experimentally obtained f -a curve, one can fix the value o f z and hence the map of relevance for the system using (14) . The fact that the same relation holds for the period doubling route in one-dimen- sional maps [ 16 ], gives added significance to the result.

In conclusion, we would like to ment ion that our method furnishes the singularity spectrum f ( a ) and the generalised dimensions Dq for a general circle map with the order of inflection z. For z = 3, t h e f - a curve has been determined earlier numerically [ 17 ]. Our method as discussed here applies to any z and is partly analytic in nature which avoids a lot of lengthy computations. The accuracy of the method can, in

Table 1 Computed values ofaqp for a few typical z values. The D~ and D_oo values obtained graphically and by calculation are also shown.

z otqp D= D_

calculated from graph calculated from graph

1.1 - 1.59502 0.936975 0.94 1.030674 1.03 2 - 1.40421 0.7089859 0.706 1.4660502 1.47 3 - 1.28857 0.6325339 0.64 1.8982408 1.8 4 - 1.23110 0.5787014 0.572 2.314572 2.4 5 - 1.19024 0.5526532 0.554 2.7637243 2.72 6 - 1.1603 0.5394935 0.53 3.2366341 3.3

50 - 1.0211 0.460973 0.458 22.11329 - 100 -1.01140 0.4399172 0.44 43.98728 -

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Volume 165, number 3 PHYSICS LETTERS A 18 May 1992

0.8

0.~

D 04

0.2

0 . 0 i i L ,

0 20 40 ~ 80

Z - +

40

3O

20 g

A i , i L

20 40 ~0 80 ~00

Z ~

1OO

b

120

Fig. 3. (a) Variation of D~(c~in) with z. (b) Variation of D-~(amax) with z.

p r inc ip le , be inc reased by cons ide r ing h igher stages

in the cons t ruc t i on o f the a t t rac to r in (9 ) . We also

address the ques t ion o f the l imi t ing b e h a v i o u r as

z--, oo. A un iversa l r e la t ion connec t ing Ctmax and O~mi n is also es tabl ished.

K M V wishes to thank the C o c h i n U n i v e r s i t y o f

Sc ience and Techno logy for the award o f a Sen io r

Resea rch Fe l lowship and G A thanks the Counc i l for

Scient i f ic and Indus t r ia l Research , N e w Delhi , for

f inanc ia l suppor t th rough a Resea rch Associa teship .

K B J acknowledges the research suppor t ex t ended by

the U G C u n d e r the Special Ass is tance Scheme.

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