- Email: [email protected]
Computable chaotic orbits
Volume 115, number 9
COMPUTABLE
PHYSICS LETTERS A
12 May 1986
CHAOTIC ORBITS
Joseph L. M c C A U L E Y Jr. 1 lnstitutt for Energiteknikk, Boks 40, N-2007 Kjel[er, Norway
and Julian I. P A L M O R E Department of Mathematics, University of Illinois, 1409 West Green Street, Urban& IL 61801, USA
Received 22 July 1985; accepted in revised form 10 March 1986
We contrast analytic properties of chaotic maps with the results of fixed-precision computation and then use Turing's ideas of computable irrational numbers to illustrate the computation of chaotic orbits to arbitrary N-bit precision. This leads to the study of chaos theory via integer maps that are automata with long-range site interactions. We also explain why the fl-shadowing lemma is not a justification for the use of fixed-precision arithmetic in chaos theory.
The motivation for our work is that deterministic chaotic systems can magnify small errors exponentially so that a change in a single digit o f an initial condition leads to a completely different orbit. In fixedprecision computation **, truncation and round-off can introduce such digit changes at every iteration o f a map so that the resulting computed orbit (pseudoorbit) is not an orbit o f the dynamical system. Rather, it is an artifact o f the chosen computational process. Three basic questions can be asked: (i) In what fundamental computational sense is the future not predictable from the past in a deterministic chaotic system? (ii) In what fundamental sense can a correctly computed orbit be chaotic? (iii) When do pseudoorbits correctly reflect the properties o f attractors at some coarse-grained level of description? We respond to questions (i) and (ii) via Turing's ideas o f computable numbers and computable functions [ 2 - 5 ] .
~ Permanent address: Physics Department, University of Houston, Houston, TX 77004. USA. ,1 In all that follows, we assume fixed-point operations [1 ]. Our analysis can be extended to include floating-point operations, but no clarity of explanation is achieved by that route. 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
We reformulate question (iii) via symbolic dynamics and explain why the/~-shadowing lemma [6] carmot be regarded as a justification for the use o f fixed-precision computations in chaos theory, as is sometimes advocated in the literature [7]. It is sufficient to discuss our ideas within the context o f one.dimensional iterated maps Xn+ 1 = f ( x n , D
)
(1)
on the unit interval 0 ~< x n ~< I, where D is a control parameter. As a "gedanken computation" one can consider a precise orbit x 0 -+ x 1 -+ ... -+ x n -+ ... where the iterates x i are typically irrational. If we change the initial data b y a small amount 8 x 0 this change is propagated according to the linearized equation 8x,,+1 ~ . f ' ( x , , , D ) S x , , .
(2)
A positive Lyapunov exponent [8] k yields deterministic chaos. With 8 x n ~ e n X S x o , the condition ,$x n "" 1 corresponds to complete loss o f information about further iterates with relaxation time n ~ - l n l S x 0 I/?,. In a mixing system [9,10] with invariant density P ( x ) , we can then u s e / ~ x ) to predict orbit statistics at some coarse-grained level b y divid433
Volume 115, number 9
PHYSICS LETTERS A
ing the unit interval into a f'mite number of bins and comparing the expected visitation frequencies with the results of computation. Mathematical analysis o f the above ideas is typically based upon the continuum of real numbers. In contrast, a computational analysis using a finite-state machine with base/a arithmetic 0a = 2, 3, 4 ...) in fLxed-point operations [1] provides one-dimensional maps with a lattice of/a N sites labeled by numbers N x = 7__?,ei//a i,
i=1
(3)
where e i = 0, 1,2 ..... /a - 1 and N defines the machine's precision (with/a = 2, N ~ 28 corresponds to "single-precision"). It follows that with finitely many lattice sites, any computed orbit must be periodic. This is the case even if the dynamical system has no periodic orbits, and amounts to an incorrect evaluation of the ftmction f. In such computations, the initial condition is exact (x 0 is given by a lattice site (3)). Truncation to the finite lattice will not mimic random physical perturbations. Indeed, it would be strange to fred periodicity as the main result o f a random process
¢2
As an example, consider the logistic map f ( x ) = D x ( 1 - x ) with D = 4 and/a = 2 on a " t o y " 6-bit computer. The Lyapunov exponent [10] is X = In 2, which means that an error in the Nth bit is propagated to the leading bit after a relaxation time n - I n 2 - N I x = N . The simplest 6-bit initial condition that does not iterate to the unstable fixed point,x = 3/4, is x 0 = 1/8. If we carry 12 bits in the multiplication register and then truncate the calculation to 6 bits, we obtain a 4-cycle after only 6 iterations: 1/8 + 7/16 63/64 -~ 3/64 ~ 11/64 + 9/16 + 63/64 + ... even though there are 26 = 64 states on the lattice. The computed orbit yields the right statistics for a division o f the unit interval into two bins 0 ~
434
12 May 1986
partitioned while 2 Nc is the cycle-length of the pseudo-orbit. In contrast with any orbit computed by fixed precision arithmetic, the orbits of the logistic map are non-periodic and truly chaotic if, beginning with a rational initial condition the arithmetic is carried out e.g., by increasing the precision to 2 n ( N - 2) + 2 in the nth iteration for the binary case discussed above. The reason for the impossibility o f periodicity is that in Xn, the 2 n ( N - 2) + 2th bit is always 1. In general, by a chaotic orbit in one dimension, we mean any bounded non-periodic orbit of a deterministic system with a positive Lyapunov exponent. In order to go systematically beyond the computation of pseudo-orbits, we will begin with the discretized picture of the continuum provided by expansions of numbers x E [0, 1] in powers o f 1//a [11], oo
X = ~-J ei//a i,
i=1
(4)
where the "digits" e i range through 0, 1, 2, ...,/a - 1. Rationals have periodic expansions in every base of arithmetic while irrationals are non-periodic in every base [11]. According to Turing [3], a computable irrational number is one whose "decimal expansion" .ele 2 ... e N can be computed to arbitrarily high but finite accuracy N by a machine. Computable numbers are numbers that are generated by algorithms [3,5] (finite length computer programs), e.g., continued fractions or ordinary square-root operations and form a countable set. Although all real numbers can be defined as limits of infinite sequences of rationals, almost all of these numbers are noncomputable in the sense that they cannot be computed via any possible algorithm. It follows from this that chaos theory, in so far as one is concerned with prediction or computation, should be built upon computable numbers, and computable functions. The computable numbers then provide the framework for understanding the sense in which deterministic orbits can be regarded as chaotic. As an example of an application of these ideas, consider the simplest chaotic map, the Bernoulli shift f ( x ) = D x mod 1. The Lyapunov exponent is given by X = In D and we will restrict to integers D = 2, 3, 4, .... If we s e t D =/a where x 0 = .6162 ... e N with e i = 0, 1, 2, ...,/a - 1, the map amounts to a base
Volume 115, number 9
PHYSICS LETTERS A
Xo= oi,I,ol, oi, o oo I oi,loool } o, ooloo oo ooo
!
F FFo o olo oI, ×o--t, o ololoo, Io x,: oooolo ,Ioo 1
x :l oooolIIo o,I - - Tape moves left
I"-
(Turing Machine)
Fig. 1. A computable chaotic orbit follows from a computable irrational initial condition via a computable map. The irrational n u m b e r x/~ - 1 = .0110101000001001... is computed by the Turing machine and used as initial data for the Bernoulli shift Xn+ t = 2Xn m od 1. The states x o, x t .... , x8 are shown to 8-bit precision. The machine has just printed e8(8) = 1. The algorithm used to compute x o is just the binary version of the usual elementary rule for obtaining a square root.
point shift one unit to the right (mod 1) o n x 0 , and with N fixed we obtain x n = 0 for n/> N. If instead we fix the last computed state at N bits, we can do this by specifying x 0 to N + n bits. One can determine the digits by rolling a fair #-sided die N + n times. This process carried out finitely many times can produce all possible f i n i t e sequences o f digits, including periodic sequences. In order to guarantee non-periodicity in the digit sequence one can instead choose an algorithm for any computable irrational number. This choice guarantees that the resulting orbit of the Bernoulli shift, with each state computed to at least N bits, will be chaotic, and is an approximation to the computable orbit corresponding to the infinite-digit irrational initial condition in the sense that the first computed bits will not be changed if the "precision" N is increased. An example is shown in fig. 1 for tt
12 May 1986
= 2 and x 0 = x / ~ - 1. It is impossible here to c o m p u t e an orbit that is more chaotic than one that is generated by a computable irrational number. In order that the iterates o f f ( x ) = tzx mod 1 define a uniform invariant density as n ~ 0% it is sufficient to choose the algorithm for a number that is normal to base/~. It is not easy to find such algorithms, but at least one is known [11] : write down the natural numbers in sequence in a given base. With/a = 2, this algorithm yields .I 1011100101110 .... It is unknown whether Vr2, zr or e is normal to any base tt [11]. Knuth [1] shows how normal numbers can be used as a basis for developing a suitable definition of infinite random sequences of digits. The randomness arises from the passage to the limit n = o% which is noncomputable. The notion of algorithmic complexity [ 13,14] has been offered as a possible explanation o f randomness of chaotic orbits [10,15]. Algorithmically complex sequences require an infinite computer program in order to def'me the number and so are by definition non-computable. Algorithmic complexity therefore cannot be used to explain chaotic properties of computed orbits. All algorithmically complex numbers are non-computable. Bernoulli shifts can be used to illustrate the weakness in the presumption that the/3-shadowing lemma has any direct implication for computation. If we consider a pseudo-orbit [6,7] x 0 ~ x 1 -+ ... ~ x n generated by making the same error a (e.g., by truncation) in the evaluation o f the Bernoulli shift, X n + 1 =/.IXn + a rood 1, then the shadowing orbit satisfies Y n + l = laYn mod 1, begins from the initial condition Y0 = x0 + a / ~ - 1) and is given by Y n = X n + a/0a - 1), so that t3 = u/0a - 1). In a fixed precision computation, the numbers x 0 and/a are rational and since a = x 1 - tzx0, it follows that a also is rational. Hence, the shadowing orbit is unstable periodic rather than chaotic. The reason why unstable periodic orbits o f Bernoulli shifts can mimic chaotic ones has been explained elsewhere [12]. In general, we expect shadowing by unstable periodic orbits, so it is the statistical properties o f these orbits rather than the chaotic orbits that are generated by fixed precision computation. I f ~ ~ 1, every chaotic orbit is trivially a shadowing orbit, but this is of no significance. Our analysis has been generalized to include Arnold's cat map [16]. Our point o f view can be generalized as follows. For an arbitrary map (I) we write 435
Volume 115, number 9 ,o
x n =i~l.= ei(n)/lai,
PHYSICS LETTERS A M
D = i=~-M ~i/lai'
(5)
where ei(n ) and 6 i can take on values 0, 1,2 .... , / a - 1 and M is an integer. The infinity o f different integer maps (p = 2, 3, 4, ...) corresponding to a given funct i o n f is a discrete analog of the arbitrariness in choice o f a basis in a vector space. A computable function f yields an algorithm that computes the digits ei(n + 1) to arbitrarily high b u t finite accuracy in terms o f the ~i and Finitely many o f the ei(n ). An example is the Bernoulli shift where ei(n + 1) = ei+l(n ). If one plots the states x 0, x 1, ..., Xn as sequences o f O's and l ' s in the spirit o f one-dimensional cellular automata, the result is an automaton that spreads backward in time (x 0 h a s N + n bits i f x n has n bits). Fig. 1 shows the uniform N-bit approximation to this result. Another example is provided by the logistic map f ( x ) = Dx(1 - x) with # = 2 and D = 4. The orbit is non-periodic and bounded with Lyapunov exponent X = In 2, hence is chaotic. In the passage to the limit n = oo, the iterates presumably are distributed according to the invariant density [10]
e ( x ) = [~x/~-O - x ) ] - l . The non-periodicity of the orbit follows from the fact that the 2n(N - 2) + 2th bit in x n is 1, with O's thereafter. One can plot the states Xo, Xl,..., x n as sequences o f O's and l ' s in the spirit o f one-dimensional cellular automata, but the "site interactions" [17] are not short-ranged. In general, our procedure transforms discrete maps o f continuous variables Xn+ 1 = f (Xn, D) into algorithms for integer variables ei(n ) that are automata [4,5] with long-ranged interactions. F r o m such examples one can see that it is the required growth of precision per iteration o f a chaotic map that limits our ability to predict the future from a deffmite initial condition, when considerations o f computation time are taken into account. The Turing machine [2,4,5 ] requires a finite number o f operations but ignores time limitations and therefore stands as a limit for computation that is analogous to the limitation placed b y the second law o f thermodynamics on real physical processes. Finally, symbolic dynamics with a division o f the unit interval into two bins o f equal size corresponds to using the first bit e l ( n ) in each iterate x n to form the binary fraction ¢n(XO) = .e I (1)e 1 (2) ... e l ( n ) . At 436
12 May 1986
this most coarse-grained level o f description the dynamics amounts to hopping between the two bins. Chaotic motion occurs if ¢n(XO) is the digit sequence of a computable irrational number. The generalization o f symbolic dynamics to 2 N bins o f equal width can be used to describe chaotic motion on freer length scales. J.I. Palrnore thanks T. Riste and the Institute for Energy Technology for the hospitality, and R. H#eghKrohn and the University of Oslo for the invitation to visit Norway. J.L. McCauley Jr. is grateful to P. Cvitanovic for a question, J. Ford for correspondence and S. Wolfram for discussions and references. He is particularly grateful to T. Riste and the Institute for Energy Technology for support and hospitality during a sabbatical. This work was supported in part b y NORDITA and the American-Scandinavian Foundation.
References [ 1] D.E. Knuth, Seminumerical algorithms, the art of computer programming, Vol. 2 (Addison-Wesley, Reading, 1981). [2] A.M. Turing, Proc. Lond. Math. Soc. (2) 42 (1937) 230. [3] A.Hodges, Alan Turing: The enigma (Simon and Schuster, 1980) pp. 91-110. [4] D. Hopkin and B. Moss, Automata (Elsevier, Amsterdam, 1976). [5] N. Cutland, Computability (Cambridge Univ. Press, Cambridge, 1980). 16] O.E. Lanford, in: Chaotic behavior of deterministic systems, eds. G. looss, R.H.G. Hellerman and R. Stora (North-Holland, Amsterdam, 1983). [7] G. Benettin, G.M. Casartelli, L. Galgani, A. Giorgilli and J.M. Strelcyn, Nuovo Cimento B 44 (1978) 183. [8] P. Collet and J.P. Eckmann, Iterated maps on the interval as dynamical systems (Birkh~iuser, Basel, 1980). [9] S. Grossmann and S. Thomae, Z. Naturforsch. a 32 (1977) 1353. [10] A.J. Lichtenberg and M.A. Lieberman, Regular and stochastic motion (Springer, Berlin, 1983). [ 11 ] I. Niven, Irrational numbers, The Carus Mathematical Monographs No. 11 (1956). [12] J.L. McCauley Jr. and J.l. Palmore, in: Scaling and disordered systems, eds. R. Pynn and A.T. Skeltorp (Plenum, New York, 1986). [13] P. Martin-Lt~f, Inf. Control 9 (1966) 602. [14] G. Chaitin, Sci. Am. 232 (1975) 47. [15] J. Ford, Phys. Today 36 (1983) 40. [16] J.l. Palmore and J.L. McCauley Jr., unpublished (1986). [17] S. Wolfram, Los Alamos Sci. 9 (1983) 2; Rev. Mod. Phys. 55 (1983) 601.
COMPUTABLE
PHYSICS LETTERS A
12 May 1986
CHAOTIC ORBITS
Joseph L. M c C A U L E Y Jr. 1 lnstitutt for Energiteknikk, Boks 40, N-2007 Kjel[er, Norway
and Julian I. P A L M O R E Department of Mathematics, University of Illinois, 1409 West Green Street, Urban& IL 61801, USA
Received 22 July 1985; accepted in revised form 10 March 1986
We contrast analytic properties of chaotic maps with the results of fixed-precision computation and then use Turing's ideas of computable irrational numbers to illustrate the computation of chaotic orbits to arbitrary N-bit precision. This leads to the study of chaos theory via integer maps that are automata with long-range site interactions. We also explain why the fl-shadowing lemma is not a justification for the use of fixed-precision arithmetic in chaos theory.
The motivation for our work is that deterministic chaotic systems can magnify small errors exponentially so that a change in a single digit o f an initial condition leads to a completely different orbit. In fixedprecision computation **, truncation and round-off can introduce such digit changes at every iteration o f a map so that the resulting computed orbit (pseudoorbit) is not an orbit o f the dynamical system. Rather, it is an artifact o f the chosen computational process. Three basic questions can be asked: (i) In what fundamental computational sense is the future not predictable from the past in a deterministic chaotic system? (ii) In what fundamental sense can a correctly computed orbit be chaotic? (iii) When do pseudoorbits correctly reflect the properties o f attractors at some coarse-grained level of description? We respond to questions (i) and (ii) via Turing's ideas o f computable numbers and computable functions [ 2 - 5 ] .
~ Permanent address: Physics Department, University of Houston, Houston, TX 77004. USA. ,1 In all that follows, we assume fixed-point operations [1 ]. Our analysis can be extended to include floating-point operations, but no clarity of explanation is achieved by that route. 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
We reformulate question (iii) via symbolic dynamics and explain why the/~-shadowing lemma [6] carmot be regarded as a justification for the use o f fixed-precision computations in chaos theory, as is sometimes advocated in the literature [7]. It is sufficient to discuss our ideas within the context o f one.dimensional iterated maps Xn+ 1 = f ( x n , D
)
(1)
on the unit interval 0 ~< x n ~< I, where D is a control parameter. As a "gedanken computation" one can consider a precise orbit x 0 -+ x 1 -+ ... -+ x n -+ ... where the iterates x i are typically irrational. If we change the initial data b y a small amount 8 x 0 this change is propagated according to the linearized equation 8x,,+1 ~ . f ' ( x , , , D ) S x , , .
(2)
A positive Lyapunov exponent [8] k yields deterministic chaos. With 8 x n ~ e n X S x o , the condition ,$x n "" 1 corresponds to complete loss o f information about further iterates with relaxation time n ~ - l n l S x 0 I/?,. In a mixing system [9,10] with invariant density P ( x ) , we can then u s e / ~ x ) to predict orbit statistics at some coarse-grained level b y divid433
Volume 115, number 9
PHYSICS LETTERS A
ing the unit interval into a f'mite number of bins and comparing the expected visitation frequencies with the results of computation. Mathematical analysis o f the above ideas is typically based upon the continuum of real numbers. In contrast, a computational analysis using a finite-state machine with base/a arithmetic 0a = 2, 3, 4 ...) in fLxed-point operations [1] provides one-dimensional maps with a lattice of/a N sites labeled by numbers N x = 7__?,ei//a i,
i=1
(3)
where e i = 0, 1,2 ..... /a - 1 and N defines the machine's precision (with/a = 2, N ~ 28 corresponds to "single-precision"). It follows that with finitely many lattice sites, any computed orbit must be periodic. This is the case even if the dynamical system has no periodic orbits, and amounts to an incorrect evaluation of the ftmction f. In such computations, the initial condition is exact (x 0 is given by a lattice site (3)). Truncation to the finite lattice will not mimic random physical perturbations. Indeed, it would be strange to fred periodicity as the main result o f a random process
¢2
As an example, consider the logistic map f ( x ) = D x ( 1 - x ) with D = 4 and/a = 2 on a " t o y " 6-bit computer. The Lyapunov exponent [10] is X = In 2, which means that an error in the Nth bit is propagated to the leading bit after a relaxation time n - I n 2 - N I x = N . The simplest 6-bit initial condition that does not iterate to the unstable fixed point,x = 3/4, is x 0 = 1/8. If we carry 12 bits in the multiplication register and then truncate the calculation to 6 bits, we obtain a 4-cycle after only 6 iterations: 1/8 + 7/16 63/64 -~ 3/64 ~ 11/64 + 9/16 + 63/64 + ... even though there are 26 = 64 states on the lattice. The computed orbit yields the right statistics for a division o f the unit interval into two bins 0 ~
434
12 May 1986
partitioned while 2 Nc is the cycle-length of the pseudo-orbit. In contrast with any orbit computed by fixed precision arithmetic, the orbits of the logistic map are non-periodic and truly chaotic if, beginning with a rational initial condition the arithmetic is carried out e.g., by increasing the precision to 2 n ( N - 2) + 2 in the nth iteration for the binary case discussed above. The reason for the impossibility o f periodicity is that in Xn, the 2 n ( N - 2) + 2th bit is always 1. In general, by a chaotic orbit in one dimension, we mean any bounded non-periodic orbit of a deterministic system with a positive Lyapunov exponent. In order to go systematically beyond the computation of pseudo-orbits, we will begin with the discretized picture of the continuum provided by expansions of numbers x E [0, 1] in powers o f 1//a [11], oo
X = ~-J ei//a i,
i=1
(4)
where the "digits" e i range through 0, 1, 2, ...,/a - 1. Rationals have periodic expansions in every base of arithmetic while irrationals are non-periodic in every base [11]. According to Turing [3], a computable irrational number is one whose "decimal expansion" .ele 2 ... e N can be computed to arbitrarily high but finite accuracy N by a machine. Computable numbers are numbers that are generated by algorithms [3,5] (finite length computer programs), e.g., continued fractions or ordinary square-root operations and form a countable set. Although all real numbers can be defined as limits of infinite sequences of rationals, almost all of these numbers are noncomputable in the sense that they cannot be computed via any possible algorithm. It follows from this that chaos theory, in so far as one is concerned with prediction or computation, should be built upon computable numbers, and computable functions. The computable numbers then provide the framework for understanding the sense in which deterministic orbits can be regarded as chaotic. As an example of an application of these ideas, consider the simplest chaotic map, the Bernoulli shift f ( x ) = D x mod 1. The Lyapunov exponent is given by X = In D and we will restrict to integers D = 2, 3, 4, .... If we s e t D =/a where x 0 = .6162 ... e N with e i = 0, 1, 2, ...,/a - 1, the map amounts to a base
Volume 115, number 9
PHYSICS LETTERS A
Xo= oi,I,ol, oi, o oo I oi,loool } o, ooloo oo ooo
!
F FFo o olo oI, ×o--t, o ololoo, Io x,: oooolo ,Ioo 1
x :l oooolIIo o,I - - Tape moves left
I"-
(Turing Machine)
Fig. 1. A computable chaotic orbit follows from a computable irrational initial condition via a computable map. The irrational n u m b e r x/~ - 1 = .0110101000001001... is computed by the Turing machine and used as initial data for the Bernoulli shift Xn+ t = 2Xn m od 1. The states x o, x t .... , x8 are shown to 8-bit precision. The machine has just printed e8(8) = 1. The algorithm used to compute x o is just the binary version of the usual elementary rule for obtaining a square root.
point shift one unit to the right (mod 1) o n x 0 , and with N fixed we obtain x n = 0 for n/> N. If instead we fix the last computed state at N bits, we can do this by specifying x 0 to N + n bits. One can determine the digits by rolling a fair #-sided die N + n times. This process carried out finitely many times can produce all possible f i n i t e sequences o f digits, including periodic sequences. In order to guarantee non-periodicity in the digit sequence one can instead choose an algorithm for any computable irrational number. This choice guarantees that the resulting orbit of the Bernoulli shift, with each state computed to at least N bits, will be chaotic, and is an approximation to the computable orbit corresponding to the infinite-digit irrational initial condition in the sense that the first computed bits will not be changed if the "precision" N is increased. An example is shown in fig. 1 for tt
12 May 1986
= 2 and x 0 = x / ~ - 1. It is impossible here to c o m p u t e an orbit that is more chaotic than one that is generated by a computable irrational number. In order that the iterates o f f ( x ) = tzx mod 1 define a uniform invariant density as n ~ 0% it is sufficient to choose the algorithm for a number that is normal to base/~. It is not easy to find such algorithms, but at least one is known [11] : write down the natural numbers in sequence in a given base. With/a = 2, this algorithm yields .I 1011100101110 .... It is unknown whether Vr2, zr or e is normal to any base tt [11]. Knuth [1] shows how normal numbers can be used as a basis for developing a suitable definition of infinite random sequences of digits. The randomness arises from the passage to the limit n = o% which is noncomputable. The notion of algorithmic complexity [ 13,14] has been offered as a possible explanation o f randomness of chaotic orbits [10,15]. Algorithmically complex sequences require an infinite computer program in order to def'me the number and so are by definition non-computable. Algorithmic complexity therefore cannot be used to explain chaotic properties of computed orbits. All algorithmically complex numbers are non-computable. Bernoulli shifts can be used to illustrate the weakness in the presumption that the/3-shadowing lemma has any direct implication for computation. If we consider a pseudo-orbit [6,7] x 0 ~ x 1 -+ ... ~ x n generated by making the same error a (e.g., by truncation) in the evaluation o f the Bernoulli shift, X n + 1 =/.IXn + a rood 1, then the shadowing orbit satisfies Y n + l = laYn mod 1, begins from the initial condition Y0 = x0 + a / ~ - 1) and is given by Y n = X n + a/0a - 1), so that t3 = u/0a - 1). In a fixed precision computation, the numbers x 0 and/a are rational and since a = x 1 - tzx0, it follows that a also is rational. Hence, the shadowing orbit is unstable periodic rather than chaotic. The reason why unstable periodic orbits o f Bernoulli shifts can mimic chaotic ones has been explained elsewhere [12]. In general, we expect shadowing by unstable periodic orbits, so it is the statistical properties o f these orbits rather than the chaotic orbits that are generated by fixed precision computation. I f ~ ~ 1, every chaotic orbit is trivially a shadowing orbit, but this is of no significance. Our analysis has been generalized to include Arnold's cat map [16]. Our point o f view can be generalized as follows. For an arbitrary map (I) we write 435
Volume 115, number 9 ,o
x n =i~l.= ei(n)/lai,
PHYSICS LETTERS A M
D = i=~-M ~i/lai'
(5)
where ei(n ) and 6 i can take on values 0, 1,2 .... , / a - 1 and M is an integer. The infinity o f different integer maps (p = 2, 3, 4, ...) corresponding to a given funct i o n f is a discrete analog of the arbitrariness in choice o f a basis in a vector space. A computable function f yields an algorithm that computes the digits ei(n + 1) to arbitrarily high b u t finite accuracy in terms o f the ~i and Finitely many o f the ei(n ). An example is the Bernoulli shift where ei(n + 1) = ei+l(n ). If one plots the states x 0, x 1, ..., Xn as sequences o f O's and l ' s in the spirit o f one-dimensional cellular automata, the result is an automaton that spreads backward in time (x 0 h a s N + n bits i f x n has n bits). Fig. 1 shows the uniform N-bit approximation to this result. Another example is provided by the logistic map f ( x ) = Dx(1 - x) with # = 2 and D = 4. The orbit is non-periodic and bounded with Lyapunov exponent X = In 2, hence is chaotic. In the passage to the limit n = oo, the iterates presumably are distributed according to the invariant density [10]
e ( x ) = [~x/~-O - x ) ] - l . The non-periodicity of the orbit follows from the fact that the 2n(N - 2) + 2th bit in x n is 1, with O's thereafter. One can plot the states Xo, Xl,..., x n as sequences o f O's and l ' s in the spirit o f one-dimensional cellular automata, but the "site interactions" [17] are not short-ranged. In general, our procedure transforms discrete maps o f continuous variables Xn+ 1 = f (Xn, D) into algorithms for integer variables ei(n ) that are automata [4,5] with long-ranged interactions. F r o m such examples one can see that it is the required growth of precision per iteration o f a chaotic map that limits our ability to predict the future from a deffmite initial condition, when considerations o f computation time are taken into account. The Turing machine [2,4,5 ] requires a finite number o f operations but ignores time limitations and therefore stands as a limit for computation that is analogous to the limitation placed b y the second law o f thermodynamics on real physical processes. Finally, symbolic dynamics with a division o f the unit interval into two bins o f equal size corresponds to using the first bit e l ( n ) in each iterate x n to form the binary fraction ¢n(XO) = .e I (1)e 1 (2) ... e l ( n ) . At 436
12 May 1986
this most coarse-grained level o f description the dynamics amounts to hopping between the two bins. Chaotic motion occurs if ¢n(XO) is the digit sequence of a computable irrational number. The generalization o f symbolic dynamics to 2 N bins o f equal width can be used to describe chaotic motion on freer length scales. J.I. Palrnore thanks T. Riste and the Institute for Energy Technology for the hospitality, and R. H#eghKrohn and the University of Oslo for the invitation to visit Norway. J.L. McCauley Jr. is grateful to P. Cvitanovic for a question, J. Ford for correspondence and S. Wolfram for discussions and references. He is particularly grateful to T. Riste and the Institute for Energy Technology for support and hospitality during a sabbatical. This work was supported in part b y NORDITA and the American-Scandinavian Foundation.
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