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Attractor dimension for the generalized Baker's transformation
Volume 99A, number 6,7
PHYSICS LETTERS
12 December 1983
ATTRACTOR DIMENSION FOR THE GENERALIZED BAKER'S TRANSFORMATION V.N. SHTERN Institute of Thermophystcs, Novosibirsk 630090, USSR Received 9 August 1983
A typical example is constructed when the metric dimension of the attractor cannot be expressed through the Lyapunov numbers
The attractor dimension is one of the main properties of dissipative dynamical systems. It may be drastmally smaller than the phase-space dimension. Therefore there is hope to reduce the system order without losing information about its attractor. Attractors of turbulent motion are usually complicated (fractal [1] ) sets having different values of the topological d T and the metric d dimensions. The values o f d T (the maximum dtmension of a manifold belonging to the attractor) is irrelevant for solving the problem of reducing the system order or finding the attractor arrangement. Contrary to d T a value o f d may be a noninteger number. There are a few definitions of noninteger dimensions [ 1]. Their analysis and relations to attractors were reported in ref. [2]. For the following self-similar examples the metric dimension may be defined by the Pontrjagin-Schnirelman [3] formula d = lira ln N ( e ) / l n ( l / e ) ,
(1)
e-*0
where N ( e ) is the minimum number of spheres of diameter e needed to cover the set. A direct use of formula (1) is not always convenient, particularly in numerical experiments. A number of formulae were suggested [4-6] which relate the attractor dimension to easier calculated Lyapunov numbers k], being the averaged principal stretching factors per unit time of an infinitesimal region of the phase space. The first and widely used notion is "the Lyapunov dimension" [4] d L = ] + ln(X 1 ... X/)/ln(1/k/+ 1), 268
(2)
where the ordering c o n v e n t i o n is h I ~> ~,2 >~ ... ~> ~-n ;]
is the maximum integer for which the numerator is not negative. Kaplan and Yorke [4] found that d L >~d and constructed examples in which d L --/:d in nongeneric (bifurcating) cases. Below are exposed unbafurcating examples with d L g=d. A well-known and simple illustration of a mixing map is the Baker transformation. Its multidimensional and dissipative generalizations were studied by Rossler [7]. We shall consider his "noodle version" namely a map of the three-dimensional cube 0 ~ x~ ~< 1 ; i = 1, 2, 3 into itself:
xt =()tixt + Pi( 1 --hi)),
)kl > 1 > X 2 > x 3 ,
X282, 5.383 < 1, Pz = ([XlXl] -- 8i[Xl Xl/8i])/(8~ -- 1), Pi = 0
if
6i =1,
X1 =5263
( ) , [ ] are fractional and integer parts of a number. The map action is shown in fig. 1 ()'1 = 6, 61 = 1, 62 = 3, 63 = 2). The initial cube is stretched along the first direction with a factor X1 and contracted along the others with factors X~-1 , X~ 1 . The produced "pencil" is cut across its long site into X1 pieces,which are placed inside the initial cube as a lattice with 82 X ~3 rows.
When the map is iterated, all points of the initial cube approach asymptotically to an attractor, which is in its construction a direct product of the interval [0,1] and Cantor sets (in the second and the third directions). Therefore a spectrum of dimensions is 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 99A, number 6,7
PHYSICS LETTERS
12 December 1983
U/A
JX~
T
Fig. 1. Noodle version of generalized Baker's transformation w~th regular lattice. (d 1 = 1, d2, d3}, where the Cantor set dimensions d i = In 8i/ln(1/3`i);i = 2, 3. Since the metric dimension is additive [ 1], it follows that d = 1 + In
62/ln(1/X2) + In 83/In(i/X3).
(3)
In general, the values fi 2, 8 3 are not expressed by Lyapunov numbers and the results o f ( 2 ) and (3) are different. The case is not bifurcational (Xi q: 1) but it is decomposable. The map and the attractor may be factorised as a product o f simple ones. On the one hand, this allows to check if the additivity condition is satisfied or not. The additivity seems natural for any concretization of a dimension notion. Besides, near every periodic point there exists its own basis (eigenvectors of the m o n o d r o m y operator). Perhaps, an attractor may be decomposed locally on this basis. In this case a number of nonzero values o f the dimension spectrum is the minimum dimension d m of manifolds, containing the attractor. (The value d m may be found as a number of linear independent vectors connecting attractor points inside an inffmitesimal ball and d T ~< d ~< dm). On the other hand, the example being decomposable is not generic enough and it is discontinuous. The map may be modified to eliminate these "defects". Letus place the image into the initial cube not as a regular lattice but as it is shown in fig. 2 (face surface, X1 = 4). Let ~21 be the image o f the initial cube after/" iterations and/)2 be a length o f projection in the second direction. The value o f 7 = P2(g21+l)/ P2(~21) does not depend o n j due to self-similarity. The image ~1 consists of 3,11 parallelepipeds with sides 1 × 3,12 X 3,I3 . For execution o f ( l ) we chose e = 3,J3,
Fig. 2. Face surface m trregular case. then N(e) = X]IN1N2, where N 1 = 1/e = X~-/(terms which do not contribute to the limit i ~ oo are neglected). If one wanted to cover ~21, then N 2 would be equal to X/2XJ but for coveting ~ in the case 7 < 1 a smaller number of elements is needed. Indeed let us produce some more iterations'to make a horizontal paraUelepiped side equal to e: 3,/~-l= 3,13(obviously, further iterations are not needed). Then X'{/P2(~2i+I)= T/X/2 a n d N 2 = 7lX123`J and after simple transformations we have d = 1 + In
8~[ln(1/X2) + In 8~/In(1/X3) ,
~ = 7/3,2,
6~ = Xl18~ •
(4)
I f the lattice is regular, then 7 = 6 23,2 and expression (4) becomes equivalent to (3). At 7 = 1 d = d L. Let, for example, 3,1 = 4, 3,2 = 1/4, X3 = 1/16 (fig. 2), then the attractor dimensions are dT=I,
d L =2,
d=1.75+0.251og2(27),
0.5~<7~< 1. To make a smooth map let us construct a transformation of a toms into itself. For this the face and back surfaces of the cube may be considered as equivalent and edges of the image components must be interattached to make the image connected and the map differentiable. Thus it is a generalization of Smale's solenoid [7,8]. The attachments may occupy a little part of the toms, so the corrections o fresults (3), (4) would be arbitrary small. 269
Volume 99A, number 6,7
PHYSICS LETTERS
T h e r e f o r e in these cases - generalized Baker's and Smale's solenoid transformations - the m e t r i c dimension o f attractors c a n n o t be expressed t h r o u g h L y a p u n o v n u m b e r s only. What gap m a y be b e t w e e n d L and d? In some cases [9] it is very small or zero. But let us consider a n-dimensional Baker's t r a n s f o r m a t i o n . R e l a t i o n (3) is easily generalized [10]. Put X1 > 1 > ~'2 > ".. > ~n, X I . X 2 ... Xn_ 1 = 1 , 8 i = 1 ifi=/=n. T h e n d L = n - 1, d = 1 + in 6 n / i n ( 1 / X n ) and d ~ 1 i f X n ~ 0. Thus the gap d L - d m a y be arbitrarity close to n - 2. The dim e n s i o n d m o f the m i n i m u m covering m a n i f o l d equals 2 in this case and d L - d m = n - 3. The a u t h o r w o u l d like to t h a n k J.D. F a r m e r for helpful discussions. He is grateful to academician
270
12 December 1983
S.S. Kutateladze and Professor M.A. Goldshtik for supporting this work. References
[ 1 ] B.B. Mandelbrot, Fractals form, chance and dxmenslon (Freeman, San Francisco, 1977). [2] J.D. Farmer, E. Ott and J.A. Yorke, Physlca 71) (1983) 153. [3] L. Pontrjagin and L. Schnirelman, Ann. Math. 33 (1932) 156. [4] J.L. Kaplan and J.A. Yorke, Lecture Notes in Mathematics, Vol. 730 (Springer, Berlin, 1979) p. 204. [5] H. Mort, Prog. Theor. Phys. 63 (1980) 1044. [6] F. Ledrappier, Commun. Math. Phys. 81 (1981) 229. [7] O.E. Rossler, Chaos and bljectlon across dimensions. [8] S. Smale, Bull. Am. Math. Soc. 73 (1967) 747. [9] J.D. Farmer, Physlca 4D (1982) 366. [10] V N. Shtern, Dokl. Akad. Nauk SSSR 270 (1983) 582.
PHYSICS LETTERS
12 December 1983
ATTRACTOR DIMENSION FOR THE GENERALIZED BAKER'S TRANSFORMATION V.N. SHTERN Institute of Thermophystcs, Novosibirsk 630090, USSR Received 9 August 1983
A typical example is constructed when the metric dimension of the attractor cannot be expressed through the Lyapunov numbers
The attractor dimension is one of the main properties of dissipative dynamical systems. It may be drastmally smaller than the phase-space dimension. Therefore there is hope to reduce the system order without losing information about its attractor. Attractors of turbulent motion are usually complicated (fractal [1] ) sets having different values of the topological d T and the metric d dimensions. The values o f d T (the maximum dtmension of a manifold belonging to the attractor) is irrelevant for solving the problem of reducing the system order or finding the attractor arrangement. Contrary to d T a value o f d may be a noninteger number. There are a few definitions of noninteger dimensions [ 1]. Their analysis and relations to attractors were reported in ref. [2]. For the following self-similar examples the metric dimension may be defined by the Pontrjagin-Schnirelman [3] formula d = lira ln N ( e ) / l n ( l / e ) ,
(1)
e-*0
where N ( e ) is the minimum number of spheres of diameter e needed to cover the set. A direct use of formula (1) is not always convenient, particularly in numerical experiments. A number of formulae were suggested [4-6] which relate the attractor dimension to easier calculated Lyapunov numbers k], being the averaged principal stretching factors per unit time of an infinitesimal region of the phase space. The first and widely used notion is "the Lyapunov dimension" [4] d L = ] + ln(X 1 ... X/)/ln(1/k/+ 1), 268
(2)
where the ordering c o n v e n t i o n is h I ~> ~,2 >~ ... ~> ~-n ;]
is the maximum integer for which the numerator is not negative. Kaplan and Yorke [4] found that d L >~d and constructed examples in which d L --/:d in nongeneric (bifurcating) cases. Below are exposed unbafurcating examples with d L g=d. A well-known and simple illustration of a mixing map is the Baker transformation. Its multidimensional and dissipative generalizations were studied by Rossler [7]. We shall consider his "noodle version" namely a map of the three-dimensional cube 0 ~ x~ ~< 1 ; i = 1, 2, 3 into itself:
xt =()tixt + Pi( 1 --hi)),
)kl > 1 > X 2 > x 3 ,
X282, 5.383 < 1, Pz = ([XlXl] -- 8i[Xl Xl/8i])/(8~ -- 1), Pi = 0
if
6i =1,
X1 =5263
( ) , [ ] are fractional and integer parts of a number. The map action is shown in fig. 1 ()'1 = 6, 61 = 1, 62 = 3, 63 = 2). The initial cube is stretched along the first direction with a factor X1 and contracted along the others with factors X~-1 , X~ 1 . The produced "pencil" is cut across its long site into X1 pieces,which are placed inside the initial cube as a lattice with 82 X ~3 rows.
When the map is iterated, all points of the initial cube approach asymptotically to an attractor, which is in its construction a direct product of the interval [0,1] and Cantor sets (in the second and the third directions). Therefore a spectrum of dimensions is 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 99A, number 6,7
PHYSICS LETTERS
12 December 1983
U/A
JX~
T
Fig. 1. Noodle version of generalized Baker's transformation w~th regular lattice. (d 1 = 1, d2, d3}, where the Cantor set dimensions d i = In 8i/ln(1/3`i);i = 2, 3. Since the metric dimension is additive [ 1], it follows that d = 1 + In
62/ln(1/X2) + In 83/In(i/X3).
(3)
In general, the values fi 2, 8 3 are not expressed by Lyapunov numbers and the results o f ( 2 ) and (3) are different. The case is not bifurcational (Xi q: 1) but it is decomposable. The map and the attractor may be factorised as a product o f simple ones. On the one hand, this allows to check if the additivity condition is satisfied or not. The additivity seems natural for any concretization of a dimension notion. Besides, near every periodic point there exists its own basis (eigenvectors of the m o n o d r o m y operator). Perhaps, an attractor may be decomposed locally on this basis. In this case a number of nonzero values o f the dimension spectrum is the minimum dimension d m of manifolds, containing the attractor. (The value d m may be found as a number of linear independent vectors connecting attractor points inside an inffmitesimal ball and d T ~< d ~< dm). On the other hand, the example being decomposable is not generic enough and it is discontinuous. The map may be modified to eliminate these "defects". Letus place the image into the initial cube not as a regular lattice but as it is shown in fig. 2 (face surface, X1 = 4). Let ~21 be the image o f the initial cube after/" iterations and/)2 be a length o f projection in the second direction. The value o f 7 = P2(g21+l)/ P2(~21) does not depend o n j due to self-similarity. The image ~1 consists of 3,11 parallelepipeds with sides 1 × 3,12 X 3,I3 . For execution o f ( l ) we chose e = 3,J3,
Fig. 2. Face surface m trregular case. then N(e) = X]IN1N2, where N 1 = 1/e = X~-/(terms which do not contribute to the limit i ~ oo are neglected). If one wanted to cover ~21, then N 2 would be equal to X/2XJ but for coveting ~ in the case 7 < 1 a smaller number of elements is needed. Indeed let us produce some more iterations'to make a horizontal paraUelepiped side equal to e: 3,/~-l= 3,13(obviously, further iterations are not needed). Then X'{/P2(~2i+I)= T/X/2 a n d N 2 = 7lX123`J and after simple transformations we have d = 1 + In
8~[ln(1/X2) + In 8~/In(1/X3) ,
~ = 7/3,2,
6~ = Xl18~ •
(4)
I f the lattice is regular, then 7 = 6 23,2 and expression (4) becomes equivalent to (3). At 7 = 1 d = d L. Let, for example, 3,1 = 4, 3,2 = 1/4, X3 = 1/16 (fig. 2), then the attractor dimensions are dT=I,
d L =2,
d=1.75+0.251og2(27),
0.5~<7~< 1. To make a smooth map let us construct a transformation of a toms into itself. For this the face and back surfaces of the cube may be considered as equivalent and edges of the image components must be interattached to make the image connected and the map differentiable. Thus it is a generalization of Smale's solenoid [7,8]. The attachments may occupy a little part of the toms, so the corrections o fresults (3), (4) would be arbitrary small. 269
Volume 99A, number 6,7
PHYSICS LETTERS
T h e r e f o r e in these cases - generalized Baker's and Smale's solenoid transformations - the m e t r i c dimension o f attractors c a n n o t be expressed t h r o u g h L y a p u n o v n u m b e r s only. What gap m a y be b e t w e e n d L and d? In some cases [9] it is very small or zero. But let us consider a n-dimensional Baker's t r a n s f o r m a t i o n . R e l a t i o n (3) is easily generalized [10]. Put X1 > 1 > ~'2 > ".. > ~n, X I . X 2 ... Xn_ 1 = 1 , 8 i = 1 ifi=/=n. T h e n d L = n - 1, d = 1 + in 6 n / i n ( 1 / X n ) and d ~ 1 i f X n ~ 0. Thus the gap d L - d m a y be arbitrarity close to n - 2. The dim e n s i o n d m o f the m i n i m u m covering m a n i f o l d equals 2 in this case and d L - d m = n - 3. The a u t h o r w o u l d like to t h a n k J.D. F a r m e r for helpful discussions. He is grateful to academician
270
12 December 1983
S.S. Kutateladze and Professor M.A. Goldshtik for supporting this work. References
[ 1 ] B.B. Mandelbrot, Fractals form, chance and dxmenslon (Freeman, San Francisco, 1977). [2] J.D. Farmer, E. Ott and J.A. Yorke, Physlca 71) (1983) 153. [3] L. Pontrjagin and L. Schnirelman, Ann. Math. 33 (1932) 156. [4] J.L. Kaplan and J.A. Yorke, Lecture Notes in Mathematics, Vol. 730 (Springer, Berlin, 1979) p. 204. [5] H. Mort, Prog. Theor. Phys. 63 (1980) 1044. [6] F. Ledrappier, Commun. Math. Phys. 81 (1981) 229. [7] O.E. Rossler, Chaos and bljectlon across dimensions. [8] S. Smale, Bull. Am. Math. Soc. 73 (1967) 747. [9] J.D. Farmer, Physlca 4D (1982) 366. [10] V N. Shtern, Dokl. Akad. Nauk SSSR 270 (1983) 582.