- Email: [email protected]
Chaotic and stochastic functions
Physica A 276 (2000) 425–440
www.elsevier.com/locate/physa
Chaotic and stochastic functions Jorge A. Gonzalez ∗ , Ramiro Pino Centro de Fsica, Instituto Venezolano de Investigaciones Cient cas, IVIC Apdo 21827, Caracas 1020-A, Venezuela Received 12 January 1999; received in revised form 1 June 1999
Abstract We study chaotic functions that are exact solutions to nonlinear maps. A generalization of these functions (which cannot be expressed as a recursive procedure anymore) can produce truly random sequences. Even if the initial conditions are known exactly, the next values are in principle unpredictable. We present a mathematical formulation of an elementary stochastic c 2000 Elsevier Science B.V. All rights reserved. process. Several applications are discussed. Keywords: Chaos; Exact solutions; Randomness; Stochastic processes; Cryptography; Disordered systems
1. Introduction The rst study of exactly solvable chaotic maps was performed by Ulam and von Neumann [1–3]. However, very recently the interest in nonlinear chaotic systems with exact solutions is increasing considerably [4 –8]. On the other hand, there exists a rushing necessity in good random number generators based in chaotic systems [9 –13]. It is not by chance that most of the above-mentioned works on exactly solvable systems (and many others) [1– 6,8–20] include important parts dedicated to stochastic processes. Indeed, there is a lack of a rigorous mathematical theory for generation of randomness [11]. In the opinion of specialists it is dicult to nd good random number generators [12]. In fact, there is no mathematical formulation of elementary stochastic process [5,11,19]. When scientists have tried to give examples of elementary random events they have had to resort to deterministic physical processes with a great uncertainty like coin tossing. Physicists have learnt quantum phenomena which are actually random, but those do not help when we are creating a mathematical model able to perform true random selections [5,11]. ∗
Corresponding author. Fax: +58-2-504-1148. E-mail addresses: [email protected] (J.A. Gonzalez), [email protected] (R. Pino)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 4 2 3 - 9
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In the present paper we investigate chaotic functions which are exact solutions to nonlinear maps. In these functions there is a critical control parameter, which is usually taken as an integer. We have generalized this parameter to be any real number. When the control parameter is fractionary the rst-return map is multivalued. For irrational numbers the rst-return map can be an erratic set of points without apparent order. We also show that there exist chaotic functions whose behavior is very similar to that presented by truly stochastic processes. In these functions there exists a “random” element that leads to an absolute unpredictability of the next value as a function of the previous values. This unpredictability is very di erent from that shown by the known chaotic systems. Finally, we discuss some applications of the obtained results. 2. Chaotic functions Ulam and von Neumann [1–3] were the rst to prove that the logistic map Xn+1 = 4Xn (1 − Xn ) ;
(1)
can be solved exactly. In fact, the function Xn = sin2 (2n )
(2)
gives us the general solution of this map. Recently, other chaotic maps have been reported to have exact solutions [4 –8]. Here we can present a generalization of these results. All the found solutions can be expressed in the following form: Xn = P(Tn ) ;
(3)
where P(t) is a periodic function (trigonometric, elliptic, hyperelliptic, Weierstrass’, etc. [4 –8]), is an integer number, T the period of the function P(t) and P(T ) de nes the initial condition. An example of exactly solvable map [8] is the following: √ Xn+1 = sin2 (z arcsin X n ) : (4) Using the transformation √ 2 (5) Yn = arcsin( X n ) ; the nonlinear map (4) can be converted into a piecewise linear map (of type Yn+1 = f(Yn ), where f(Yn ) can be de ned for di erent intervals of Yn by linear functions). Thus, applying the well-known formula N 1 X df(Yn ) ln ; (6) = lim N →∞ N dYn n=0
it is possible to obtain an exact analytic expression for the Lyapunov exponent = ln z :
(7)
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Fig. 1. First-return maps produced by chaotic function (9) for z integer. (a) z = 2; (b) z = 3.
For z ¿ 1 the map (4) is chaotic. A similar procedure can be applied to all the maps for which the functions (3) are general solutions. In general, = ln :
(8)
After simple algebra it can be shown that the function Xn = sin2 (z n )
(9)
represents the exact general solution to the map (4) when z is an integer. In fact, the rst return maps shown in Fig. 1 (and the corresponding time series) obtained from the map (4), for z integer, can be reproduced by the function (9). However, for a fractionary z the dynamics contained in function (9) is quite di erent from that of map (4). Indeed, for a fractionary z ¿ 1, the rst-return chaotic map generated by (9) is multivalued (see Figs. 2 and 3).
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Fig. 2. Complex dynamics described by function (9) with z = 32 . (a) Two-valued rst-return map; (b) Time series.
Let z be a rational number expressed as z=
p ; q
(10)
where p and q are relative prime numbers. Then the rst-return map produced by function (9) is a curve such that, in general, for a value of Xn we will have q values of Xn+1 . On the other hand, for a value of Xn+1 we will have p values of Xn . Actually, these curves are Lissajous curves [8]. We just should note that the time-series Xn is chaotic and unpredictable (see Fig. 2), unless we know the exact value of . Consider the r-dimensional map Xn+1 = g(Xn ; Xn−1 ; : : : ; Xn−r+1 ) :
(11)
Even when the map (11) is chaotic and the error in the initial condition will increase exponentially the value Xn+1 is always calculable if we know the previous r values.
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Fig. 3. Multivalued rst-return maps generated by function (9). (a) z = 43 ; (b) z = 74 ; (c) z = 65 .
After a rigorous analysis of function (9) we arrive at surprising conclusions. For a fractionary z ¿ 1, the function is not only chaotic, but its next value is impossible to predict (from the previous values) unless we know exactly. When z is integer, the initial condition X0 de nes univocally the value of (any value of out of the interval 0 ¡ ¡ 1 de ning X0 is equivalent to one in that interval). If z is fractionary this is not so, there exists an in nite number of values of that satisfy the initial conditions.
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The time-series produced for di erent values of satisfying the initial condition is di erent in most of the cases. Let us consider the case of z = 32 (see Fig. 2). The rst-return map is two-valued and can be de ned parametrically in the following way: Xn = 1 − t 2 ;
(12)
Xn+1 = 12 (1 + t)(1 − 2t)2 ;
(13)
where −1 ¡ t ¡ 1. If we wish to calculate Xn+1 from the value Xn we will have two choices: Xn+1 = 12 [1 ± (1 − 4Xn )(1 − Xn )1=2 ] :
(14)
The value Xn+1 could be expressed as a well-de ned function of the previous values if (1 − Xn )1=2 could be a rational function of the previous values. However, each time we try to do this we meet the same diculty because the previous values are also irrational functions of the past values: Xn = 12 [1 ± (1 − 4Xn−1 )(1 − Xn−1 )1=2 ] :
(15)
This process can continue to in nity. A di erent way to see this phenomenon is the following. Consider the family of functions: Xnk = sin2 [(0 + k)z n ] ;
(16)
where = 0 + k, k is integer. For all k, the time series Xnk (k xed, n as time) have the same initial condition. However, for z ¿ 1 (fractionary) all the time series are di erent. That is why the period of function Xnk (now n is xed and we take k as variable) is di erent for di erent n. That is, Xn+1 cannot be determined by Xn . Moreover, Xn+1 cannot be determined by any number of previous values. Let us see the following simple example with z = 32 . Suppose Xn = 0. Now we have two possibilities Xn+1 = 0 or Xn+1 = 1. Assume 0 = 0 (n = 0). For any = k (k integer), Xn = 0. Now, Xn+1 = sin2 [(3=2)k]. So, Xn+1 = 0 for k even, and Xn+1 = 1 for k odd. But there is no way we can know k from the statement Xn = 0 (for all k integer this statement is true). This happens for all points Xn except Xn = 14 and Xn = 1. But these two points are a set of zero measure. That is, for almost all the points in the interval 0 ¡ Xn ¡ 1 the next value is unpredictable. For z irrational there are in nite possibilities for Xn+1 . All values are unpredictable. But let us continue with the simpler case z = 32 . Suppose now that = 2m , where m is integer. Note that in this case X0 = 0. But, unless we know , we never will know when the value Xn+1 will be equal to 1. We can have a string of m + 1 zeros (m can be as large as we want) and only in the point Xm+1 = 1 the sequence changes from a string of zeros to the value 1. For instance, take m = 3. Then we have X0 = 0; X1 = 0; X2 = 0; X3 = 0. Now, after four values of the sequence we have still the uncertainty about the next value. In fact, in this case X4 = 1. Nevertheless, if m = 5, we can still have two more zeros before a sequence changes to have the value 1. So, for any nite number m + 1 of
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Fig. 4. The rst-return map produced by function (9) with z = e (e is the base of natural logarithms, i.e.,√an irrational number) is a set of points with no apparent order (a), and after the transformation Y =2= arcsin X (b).
previous values, X0 ; X1 ; X2 ; : : : ; Xn ; the next value is not de ned by the previous values. Note that in this example we can have a string of zeros, but this is because the value Xn = 0 is a “pseudo xed point” of the map (Xn ; Xn+1 ) due to the intersection of the graph in Fig. 2 with the line Xn+1 = Xn . In general, the sequence is very chaotic. On the other hand, the uncertainty about which is the next value remains for all the points in the interval 0 ¡ Xn ¡ 1 except for Xn = 14 and Xn = 1 (see Fig. 2 where the image of these points is uniquely de ned). The general uncertainty increases for p ¿ q ¿ 2, in this case the unpredictability is true for all values of Xn . On the other hand, if z is irrational, then the points on the rst-return map (Xn ; Xn+1 ) will ll the square 06Xn 61, 06Xn+1 61 (see Fig. 4(a)). For a large but nite number n, the map is an erratic set of points. For z/1 (irrational) the behavior of the time series is very similar to that produced by noisy systems which have no apparent order. However, we should say that
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function (9) with some classes of irrational z produces sequences that can be expressed as maps of type (11). For example, for algebraic irrationals of type z = m1=k ;
(17)
where m and k are integers, the sequence generated by the help of function (9) is a solution of the map √ (18) Xn+k = sin2 (m arcsin X n ) ; where the function Xn+k = f(Xn ) is a polynomial. Note that in this case, we should de ne the rst k values as initial conditions in order to obtain the complete sequence. It is interesting to remark that a sequence with z = p=q, (p ¿ q ¿ 1) cannot be expressed as a map of type (11). Therefore, this sequence is more random than that produced by an irrational number of type z = m1=k . In the case of these algebraic numbers, the map (Xn ; Xn+1 ) lls the plane, but the map (Xn ; Xn+k ) is a one-valued function. However, even these sequences are very important since in practice we work with nite sequences and making k of the order of the size of the sample, leads to the unpredictability of the nite sequence. Now, we should say some words about the use of these functions in actual numerical calculations. The argument of function (9) increases exponentially. So, there can be some problems in generating very large sequences. A practical solution is to change parameter after a xed number n=N of sequences values Xn . Suppose N is a number for which there are not calculation problems. We can determine a new value of the parameter after each set of N values of Xn , in such a way that the last value of Xn is obtained with the new . For producing the new set of values of Xn we start with n = 0. This procedure can be repeated the desired number of times (remember that even if the sequence is nite, it will be unpredictable; and a sequence formed as a set of unpredictable sequences will be also unpredictable). It can be shown that there exists always a such that, with it, the original function will produce the same sequence as that generated with the procedure of changing . Hence, the dynamical and statistical properties of the series obtained changing coincide with those of the original one. Nevertheless, the following considerations are the most important about the generation of random numbers. For the calculation of truly random numbers with function (9) the best way is to use an irrational z. This irrational z has not to be a large number. For instance, we can use z = e. The geometrical place of the return map for z irrational is the whole square 06Xn 61; 06Xn+1 61. So, we do not have to worry about the method for determining the next value of . For example, we can use the following method in order to change parameter after each set of N sequence values. Let us de ne s = AWs ;
(19)
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where Ws is produced by a chaotic map of the form Ws+1 = f(Ws ); s is the order number of in such a way that s = 1 corresponds to the used for the rst set of N values Xn ; s = 2 for the second set, etc. In this context, Eq. (4) can play the role of the chaotic map. The magnitude A can be chosen as a not too large number. However, the inequality A ¿ 1 should hold in order to keep the absolute unpredictability. Another important question about good random numbers is to have a generator able to produce uniformly distributed points. After that, other distributions can be obtained [21–23]. The points generated by function (9) are truly random in the sense that they are independent, but they are not uniformly distributed. Their distribution is peaked around X = 0 and 1. In fact, the analysis shows that the probability density P(X ) behaves as 1 P(X ) ≈ p : (20) X (1 − X ) By means of the transformation of Eq. (5) we can obtain random numbers uniformly distributed on the interval (0; 1). Note that this is not a trivial pseudorandom number generator because the numbers Xn were already unpredictable. Fig. 4(b) shows the map (Yn ; Yn+1 ) constructed after transformation (5). Note that the points are uniformly distributed. Moreover, applying the transformation (5) to the sequence generated with the scheme of changing parameter , we obtain uniform random numbers in the same way as with the original time series. We have performed several statistical tests with the functions (9) (after the transformation (5)). Among them are the following: the Central Limit Theorem test, the moments calculation, the variance calculation and the 2 test. The sequence Yn has passed all these tests satisfactorily. With the help of these tests we can verify that {Yn } is a set of mutually independent variables. Now, let us introduce the auto-correlation function: C(m) = hXi Xi+m i − hXi i2 ;
(21)
where h i is an average over all i with i = 1; 2; : : : ; N [24]. The function C(m) measures the correlation between the numbers placed on two sites that are separated by m steps. First, we wish to apply this function to the chaotic map (4) for z = 1:7 (see Fig. 5). In general, for the known chaotic maps, |C(m)| decays with m. Note that there is a range of this dependence that is related to the correlation or memory time. As a rule, if we increase the Lyapunov exponent, the correlation time tends to decrease. However, if we calculate the auto-correlation function for the sequence Yn (using Eqs. (9) and (5)) with z = , then we obtain that the correlation time is exactly zero. Already for m = 1 there are no correlations. In Refs. [25 –27] a new method is developed, which allows to compare the randomness of di erent sequences of equal length. In these works, a measure of randomness (we will call it R) is introduced.
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√ Fig. 5. Autocorrelation function for the map Xn+1 = sin2 (z arcsin X n ) with z = 1:7.
Suppose we have a sequence of values U1 ; U2 ; U3 ; : : : ; UN . Form a sequence of vectors X (i) = [Ui ; Ui+1 ; : : : ; Ui+m−1 ] :
(22)
Now, we will de ne some variables: Cim (r) = (number of j such that d[X (i); X (j)]6r)=(N − m + 1) ;
(23)
where d[X (i); X (j)] is the distance between two vectors, which is de ned as follows: d[X (i); X (j)] = max(|Ui+k−1 − Uj+k−1 |);
(k = 1; 2; : : : ; m) :
(24)
Another important quantity is m (r) =
N −m+1 X
ln Cim (r)=(N − m + 1) :
(25)
i=1
From m (r), we de ne R(m; r; N ) = m (r) − m+1 (r) :
(26)
This measure depends on the resolution parameter r and an “embedding” parameter m. The calculation of R allows to address the system randomness. This technique has been proved to be very e ective in determining system complexity [25 –27]. For given r and m we have a maximum possible randomness. A sequence with maximum randomness is uncorrelated. The randomness generated by our function (9) with z irrational is the maximum possible for the given r and m. The same is obtained for the sequence generated with the scheme of changing parameter . For instance, if r = 0:025, the maximum possible randomness is R = ln 40. The randomness of function (9) with z = approaches the value R = 3:688 for increasing N . For comparison, the
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randomness of the logistic map at the point of full chaos is R = 0:693. Even if we further decrease r and increase m and N , for the logistic map (and other usually chaotic maps), R saturates and remains constant. On the other hand, for r → 0, the randomness of function (9) with z = tends to the maximum possible value, i.e. R → ln(1=r). In the same framework, we can de ne the correlation dimension [28]: ln C m (r) ; (27) r→0 N →∞ ln r PN −m+1 m Ci (r)=(N − m + 1). Here m plays the role of embedding where C m (r) = i=1 dimension. For completely random systems the correlation dimension is equal to the embedding dimension. When we calculate m for function (9) with z = , we nd the relation m = m. In comparison, for the known chaotic systems: m = lim lim
lim m = ;
m→∞
where is a nite number. For suciently large m; m tends to , where .m. Now, we resort to a very important method developed in the papers [29 –31], in order to investigate random and chaotic systems. This technique is very powerful in distinguishing chaos from random time series. The idea of the method is the following. One can make short-term predictions that are based on a library of past patterns in a time series (the method of nonlinear forecasting is described in Refs. [29 –31] and the references quoted therein). By comparing the predicted and actual values, one can make distinctions between random sequences and deterministic chaos. For chaotic (but correlated) time series the accuracy of the nonlinear forecast falls o with increasing prediction–time interval. On the other hand, for truly random sequences, the forecasting accuracy is independent of the prediction interval. If the sequence values are correlated, then future values may, to some extent, be predicted from the behavior of past values that are similar to those of the present. For uncorrelated random sequences the error remains constant. The prediction accuracy is measured by the coecient of correlation between predicted and observed values. For deterministic chaotic sequences this coecient falls as predictions extend further into the future. Suppose we have a sequence U1 ; U2 ; : : : ; UN . Now, we construct a map with the dependence of Un (predicted) as a “function” of Un (observed). If we have a correlated chaotic sequence, this dependence is almost a straight line, i.e. Un (predicted) ≈ Un (observed) (when the forecasting method is applied for one time step into the future). When we increase the number of time steps into the future, this relation deteriorates. The decrease with time of the correlation coecient between predicted and actual values has been used to calculate the largest positive Lyapunov exponent of a time series [30].
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Fig. 6. Predicted values (YnP ) one time step into the future versus observed values (YnO ) for function (9) after the transformation (5) (with z = ).
We have applied this method of investigation to our function (9) after the transformation (5), with z = . Fig. 6shows the predicted values one time step into the future versus observed values. The same picture is obtained for any number m of time steps into the future. As expected, the correlation coecient is independent of the prediction time. Even when the method is applied with a prediction–time interval m = 1, the correlation coecient is zero. This shows that the corresponding time series behaves as a random sequence. We should say that the pseudorandom number generators described in Ref. [10] can pass some of the statistical tests devised to check pseudorandomness [9,11]. However, hidden errors in these generators have been found [10]. Ferrenberg et al. [10] traced the errors to the dependence in the random numbers. Indeed, they are all based on recursive algorithms. We believe that the strongest result of the present paper is the theoretical proof (discussed between Eqs. (11) and (17)) that the outcomes of function (9) are independent. 3. Applications 3.1. Stochastic processes There exists the belief that chaotic processes can be de ned by algorithms, but random processes cannot. On the other hand, it is important to construct pseudo-random number generators and stochastic processes from chaotic systems.
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Chaotic maps have been used as pseudorandom number generators for obtaining samples of variable with di erent distributions [5,11]. Many known random number generators are based on chaotic maps as the following Xn+1 = (aXn + b) mod(T ) :
(28)
Sometimes the very logistic map is used with this aim [5,11,15]. Behind this e ort is the concept that “pseudorandomness is chaos with a very large Lyapunov exponent”. How could actual randomness be de ned. Suppose we have an ideal generator for truly random numbers. In this case no matter how many numbers we have generated, the value of the next number will be still unknown. That is, there is no way to write down a formula that will give the value of the next number in terms of the previous numbers, no matter how many numbers we already have. No recursive algorithm exists that will generate the sequence. The paradigm of random process is the coin toss. But this is not a mathematical procedure that would enable us to make investigations. A chaotic dynamical system as the map (11) can generate a sequence that resemble one generated by an ideal random process. But, given the same initial condition we get the same sequence, unlike a real random process. It is very important to have a process that could be de ned as a standard random process. We think that a practical way out is to use the chaotic functions we are investigating in this paper. We will introduce the concept of a-random sequences which will be free of the limitations of the philosophically inspired de nitions reviewed in Refs. [5,11]. Let us de ne Xn as an a-random sequence if it cannot be a solution of a map of type (11). Hence, there is no recursive algorithm for the generation of an a-random sequence. The ideal random coin toss cannot be expressed as a map of type (11), so it is an a-random sequence. Do we know explicit mathematical functions that represent a-random sequences? The answer is yes! Function (9) for z ¿ 1 fractionary is an a-random sequence. We can use the chaotic function of (9) to model stochastic processes on a computer. 3.2. Monte Carlo method The specialists in Monte Carlo method [9,11] face the problem of testing pseudorandom number generators. How to create a sequence of numbers indistinguishable from those produced by a truly random process. This question possesses practical importance because a very large number of calculations are performed using the Monte Carlo method with pseudorandom numbers. Almost all known generators (in some cases) give rise to incorrect results because they deviate from randomness [10]. There is a lack of mathematical basis for generation of randomness [9,11]. Most of the algorithms used in Monte Carlo calculations are quite predictable. There is also the quasi-Monte Carlo method [11]. The advantage of this method is that the generated points cover the phase space very uniformly and the numerical
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integration has very fast convergence rates. However this algorithm is unappropiate for many Monte Carlo calculations because the generated points are very regular and predictable. The chaotic functions studied here present the required randomness and the generated points cover the space uniformly. 3.3. Cryptography It can be shown [5] that many of the widely used cryptosystems [32,33] are equivalent to chaotic maps. These maps are used to generate a sequence of numbers to be utilized as codes for letters. It does not matter how chaotic a nonlinear map can be, it is not truly random. A text encoded using function (9) cannot be decoded even if we unveil the procedure (but remains unknown). If you do not know it, you know nothing. On the other hand, it is easy to compress and transmit the encoded information. 3.4. Theoretical calculations Functions (9) have also another important application. They are relevant by themselves as theoretical paradigms of stochastic processes. Considering the fact that these are explicit functions, we can use them to solve (analytically) many theoretical problems in stochastic dynamical systems. 4. Discussion and conclusions In the present paper we have studied the chaotic functions Xn = sin2 (z n ) (Eq. (9)). For integer z, these functions represent the general solutions to chaotic maps. The Lyapunov exponent of these maps can be calculated exactly. However, when z is fractionary the produced rst-return map is multivalued. Moreover, if z is irrational the produced rst-return map is an erratic set of points very similar to that produced by noisy systems. Actually, for a fractionary z, it can be shown that function (9) describes a complex dynamics which is in principle unpredictable using the previous values (in this sense, these sequences are di erent from those appearing in the known chaotic systems, where the “unpredictability” is the result of the sensitive dependence on initial conditions). The sequences produced by these functions cannot be generated by a nite recursive algorithm. They are deterministic only in the sense that they are generated by an explicit function. However, there does not exist a map of type (11) to which they are solutions. Therefore, for any nite number r of previous values (Xn ; Xn−1 ; : : : ; Xn−r+1 ), it is impossible to de ne the value Xn+1 as a function of the former. Hence, given r initial values of Xn , the next value is undecidable. There exist in nite values of (in function (9)) that would give the same r initial values of Xn , which, nevertheless, produce
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di erently proceeding sequences. There is no nite method to determine the particular that produced that particular sequence. Thus, there is no method to determine the next value. Several authors [5,11,19] have discussed the di erences between a chaotic sequence and a truly random sequence. The question about the existence of nonrecursive algorithm is fundamental. Here we introduced the concept of a-random sequences to de ne those sequences that can be produced by a determinist algorithm, but not by a recursive algorithm. In fact, function (9), with z ¿ 1 fractionary, satisfy all the known tests for randomness [9]. We have discussed several applications of the obtained results which include: stochastic processes, random number generators, Monte Carlo method, cryptography and theoretical calculations (of course this list could be widely extended). Finally, having in mind that there is no rigorous operational de nition of randomness in terms of classical mathematical primitives [5,11,19], we believe that the introduction of the dynamics generated by function (9) as elementary random process will be very useful and it can serve as a standard for the study of other complex dynamics. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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440 [29] [30] [31] [32] [33]
J.A. Gonzalez, R. Pino / Physica A 276 (2000) 425–440 G. Sugihara, R.M. May, Nature 344 (1990) 734. D.J. Wales, Nature 350 (1991) 485. A.A. Tsonis, J.B. Elsner, Nature 358 (1992) 217. D. Denning, Cryptography and Data Security, Addison-Wesley, Reading, 1982. B. Schneier, Applied Cryptography, Wiley, New York, 1994.
www.elsevier.com/locate/physa
Chaotic and stochastic functions Jorge A. Gonzalez ∗ , Ramiro Pino Centro de Fsica, Instituto Venezolano de Investigaciones Cient cas, IVIC Apdo 21827, Caracas 1020-A, Venezuela Received 12 January 1999; received in revised form 1 June 1999
Abstract We study chaotic functions that are exact solutions to nonlinear maps. A generalization of these functions (which cannot be expressed as a recursive procedure anymore) can produce truly random sequences. Even if the initial conditions are known exactly, the next values are in principle unpredictable. We present a mathematical formulation of an elementary stochastic c 2000 Elsevier Science B.V. All rights reserved. process. Several applications are discussed. Keywords: Chaos; Exact solutions; Randomness; Stochastic processes; Cryptography; Disordered systems
1. Introduction The rst study of exactly solvable chaotic maps was performed by Ulam and von Neumann [1–3]. However, very recently the interest in nonlinear chaotic systems with exact solutions is increasing considerably [4 –8]. On the other hand, there exists a rushing necessity in good random number generators based in chaotic systems [9 –13]. It is not by chance that most of the above-mentioned works on exactly solvable systems (and many others) [1– 6,8–20] include important parts dedicated to stochastic processes. Indeed, there is a lack of a rigorous mathematical theory for generation of randomness [11]. In the opinion of specialists it is dicult to nd good random number generators [12]. In fact, there is no mathematical formulation of elementary stochastic process [5,11,19]. When scientists have tried to give examples of elementary random events they have had to resort to deterministic physical processes with a great uncertainty like coin tossing. Physicists have learnt quantum phenomena which are actually random, but those do not help when we are creating a mathematical model able to perform true random selections [5,11]. ∗
Corresponding author. Fax: +58-2-504-1148. E-mail addresses: [email protected] (J.A. Gonzalez), [email protected] (R. Pino)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 4 2 3 - 9
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In the present paper we investigate chaotic functions which are exact solutions to nonlinear maps. In these functions there is a critical control parameter, which is usually taken as an integer. We have generalized this parameter to be any real number. When the control parameter is fractionary the rst-return map is multivalued. For irrational numbers the rst-return map can be an erratic set of points without apparent order. We also show that there exist chaotic functions whose behavior is very similar to that presented by truly stochastic processes. In these functions there exists a “random” element that leads to an absolute unpredictability of the next value as a function of the previous values. This unpredictability is very di erent from that shown by the known chaotic systems. Finally, we discuss some applications of the obtained results. 2. Chaotic functions Ulam and von Neumann [1–3] were the rst to prove that the logistic map Xn+1 = 4Xn (1 − Xn ) ;
(1)
can be solved exactly. In fact, the function Xn = sin2 (2n )
(2)
gives us the general solution of this map. Recently, other chaotic maps have been reported to have exact solutions [4 –8]. Here we can present a generalization of these results. All the found solutions can be expressed in the following form: Xn = P(Tn ) ;
(3)
where P(t) is a periodic function (trigonometric, elliptic, hyperelliptic, Weierstrass’, etc. [4 –8]), is an integer number, T the period of the function P(t) and P(T ) de nes the initial condition. An example of exactly solvable map [8] is the following: √ Xn+1 = sin2 (z arcsin X n ) : (4) Using the transformation √ 2 (5) Yn = arcsin( X n ) ; the nonlinear map (4) can be converted into a piecewise linear map (of type Yn+1 = f(Yn ), where f(Yn ) can be de ned for di erent intervals of Yn by linear functions). Thus, applying the well-known formula N 1 X df(Yn ) ln ; (6) = lim N →∞ N dYn n=0
it is possible to obtain an exact analytic expression for the Lyapunov exponent = ln z :
(7)
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Fig. 1. First-return maps produced by chaotic function (9) for z integer. (a) z = 2; (b) z = 3.
For z ¿ 1 the map (4) is chaotic. A similar procedure can be applied to all the maps for which the functions (3) are general solutions. In general, = ln :
(8)
After simple algebra it can be shown that the function Xn = sin2 (z n )
(9)
represents the exact general solution to the map (4) when z is an integer. In fact, the rst return maps shown in Fig. 1 (and the corresponding time series) obtained from the map (4), for z integer, can be reproduced by the function (9). However, for a fractionary z the dynamics contained in function (9) is quite di erent from that of map (4). Indeed, for a fractionary z ¿ 1, the rst-return chaotic map generated by (9) is multivalued (see Figs. 2 and 3).
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Fig. 2. Complex dynamics described by function (9) with z = 32 . (a) Two-valued rst-return map; (b) Time series.
Let z be a rational number expressed as z=
p ; q
(10)
where p and q are relative prime numbers. Then the rst-return map produced by function (9) is a curve such that, in general, for a value of Xn we will have q values of Xn+1 . On the other hand, for a value of Xn+1 we will have p values of Xn . Actually, these curves are Lissajous curves [8]. We just should note that the time-series Xn is chaotic and unpredictable (see Fig. 2), unless we know the exact value of . Consider the r-dimensional map Xn+1 = g(Xn ; Xn−1 ; : : : ; Xn−r+1 ) :
(11)
Even when the map (11) is chaotic and the error in the initial condition will increase exponentially the value Xn+1 is always calculable if we know the previous r values.
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Fig. 3. Multivalued rst-return maps generated by function (9). (a) z = 43 ; (b) z = 74 ; (c) z = 65 .
After a rigorous analysis of function (9) we arrive at surprising conclusions. For a fractionary z ¿ 1, the function is not only chaotic, but its next value is impossible to predict (from the previous values) unless we know exactly. When z is integer, the initial condition X0 de nes univocally the value of (any value of out of the interval 0 ¡ ¡ 1 de ning X0 is equivalent to one in that interval). If z is fractionary this is not so, there exists an in nite number of values of that satisfy the initial conditions.
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The time-series produced for di erent values of satisfying the initial condition is di erent in most of the cases. Let us consider the case of z = 32 (see Fig. 2). The rst-return map is two-valued and can be de ned parametrically in the following way: Xn = 1 − t 2 ;
(12)
Xn+1 = 12 (1 + t)(1 − 2t)2 ;
(13)
where −1 ¡ t ¡ 1. If we wish to calculate Xn+1 from the value Xn we will have two choices: Xn+1 = 12 [1 ± (1 − 4Xn )(1 − Xn )1=2 ] :
(14)
The value Xn+1 could be expressed as a well-de ned function of the previous values if (1 − Xn )1=2 could be a rational function of the previous values. However, each time we try to do this we meet the same diculty because the previous values are also irrational functions of the past values: Xn = 12 [1 ± (1 − 4Xn−1 )(1 − Xn−1 )1=2 ] :
(15)
This process can continue to in nity. A di erent way to see this phenomenon is the following. Consider the family of functions: Xnk = sin2 [(0 + k)z n ] ;
(16)
where = 0 + k, k is integer. For all k, the time series Xnk (k xed, n as time) have the same initial condition. However, for z ¿ 1 (fractionary) all the time series are di erent. That is why the period of function Xnk (now n is xed and we take k as variable) is di erent for di erent n. That is, Xn+1 cannot be determined by Xn . Moreover, Xn+1 cannot be determined by any number of previous values. Let us see the following simple example with z = 32 . Suppose Xn = 0. Now we have two possibilities Xn+1 = 0 or Xn+1 = 1. Assume 0 = 0 (n = 0). For any = k (k integer), Xn = 0. Now, Xn+1 = sin2 [(3=2)k]. So, Xn+1 = 0 for k even, and Xn+1 = 1 for k odd. But there is no way we can know k from the statement Xn = 0 (for all k integer this statement is true). This happens for all points Xn except Xn = 14 and Xn = 1. But these two points are a set of zero measure. That is, for almost all the points in the interval 0 ¡ Xn ¡ 1 the next value is unpredictable. For z irrational there are in nite possibilities for Xn+1 . All values are unpredictable. But let us continue with the simpler case z = 32 . Suppose now that = 2m , where m is integer. Note that in this case X0 = 0. But, unless we know , we never will know when the value Xn+1 will be equal to 1. We can have a string of m + 1 zeros (m can be as large as we want) and only in the point Xm+1 = 1 the sequence changes from a string of zeros to the value 1. For instance, take m = 3. Then we have X0 = 0; X1 = 0; X2 = 0; X3 = 0. Now, after four values of the sequence we have still the uncertainty about the next value. In fact, in this case X4 = 1. Nevertheless, if m = 5, we can still have two more zeros before a sequence changes to have the value 1. So, for any nite number m + 1 of
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Fig. 4. The rst-return map produced by function (9) with z = e (e is the base of natural logarithms, i.e.,√an irrational number) is a set of points with no apparent order (a), and after the transformation Y =2= arcsin X (b).
previous values, X0 ; X1 ; X2 ; : : : ; Xn ; the next value is not de ned by the previous values. Note that in this example we can have a string of zeros, but this is because the value Xn = 0 is a “pseudo xed point” of the map (Xn ; Xn+1 ) due to the intersection of the graph in Fig. 2 with the line Xn+1 = Xn . In general, the sequence is very chaotic. On the other hand, the uncertainty about which is the next value remains for all the points in the interval 0 ¡ Xn ¡ 1 except for Xn = 14 and Xn = 1 (see Fig. 2 where the image of these points is uniquely de ned). The general uncertainty increases for p ¿ q ¿ 2, in this case the unpredictability is true for all values of Xn . On the other hand, if z is irrational, then the points on the rst-return map (Xn ; Xn+1 ) will ll the square 06Xn 61, 06Xn+1 61 (see Fig. 4(a)). For a large but nite number n, the map is an erratic set of points. For z/1 (irrational) the behavior of the time series is very similar to that produced by noisy systems which have no apparent order. However, we should say that
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function (9) with some classes of irrational z produces sequences that can be expressed as maps of type (11). For example, for algebraic irrationals of type z = m1=k ;
(17)
where m and k are integers, the sequence generated by the help of function (9) is a solution of the map √ (18) Xn+k = sin2 (m arcsin X n ) ; where the function Xn+k = f(Xn ) is a polynomial. Note that in this case, we should de ne the rst k values as initial conditions in order to obtain the complete sequence. It is interesting to remark that a sequence with z = p=q, (p ¿ q ¿ 1) cannot be expressed as a map of type (11). Therefore, this sequence is more random than that produced by an irrational number of type z = m1=k . In the case of these algebraic numbers, the map (Xn ; Xn+1 ) lls the plane, but the map (Xn ; Xn+k ) is a one-valued function. However, even these sequences are very important since in practice we work with nite sequences and making k of the order of the size of the sample, leads to the unpredictability of the nite sequence. Now, we should say some words about the use of these functions in actual numerical calculations. The argument of function (9) increases exponentially. So, there can be some problems in generating very large sequences. A practical solution is to change parameter after a xed number n=N of sequences values Xn . Suppose N is a number for which there are not calculation problems. We can determine a new value of the parameter after each set of N values of Xn , in such a way that the last value of Xn is obtained with the new . For producing the new set of values of Xn we start with n = 0. This procedure can be repeated the desired number of times (remember that even if the sequence is nite, it will be unpredictable; and a sequence formed as a set of unpredictable sequences will be also unpredictable). It can be shown that there exists always a such that, with it, the original function will produce the same sequence as that generated with the procedure of changing . Hence, the dynamical and statistical properties of the series obtained changing coincide with those of the original one. Nevertheless, the following considerations are the most important about the generation of random numbers. For the calculation of truly random numbers with function (9) the best way is to use an irrational z. This irrational z has not to be a large number. For instance, we can use z = e. The geometrical place of the return map for z irrational is the whole square 06Xn 61; 06Xn+1 61. So, we do not have to worry about the method for determining the next value of . For example, we can use the following method in order to change parameter after each set of N sequence values. Let us de ne s = AWs ;
(19)
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where Ws is produced by a chaotic map of the form Ws+1 = f(Ws ); s is the order number of in such a way that s = 1 corresponds to the used for the rst set of N values Xn ; s = 2 for the second set, etc. In this context, Eq. (4) can play the role of the chaotic map. The magnitude A can be chosen as a not too large number. However, the inequality A ¿ 1 should hold in order to keep the absolute unpredictability. Another important question about good random numbers is to have a generator able to produce uniformly distributed points. After that, other distributions can be obtained [21–23]. The points generated by function (9) are truly random in the sense that they are independent, but they are not uniformly distributed. Their distribution is peaked around X = 0 and 1. In fact, the analysis shows that the probability density P(X ) behaves as 1 P(X ) ≈ p : (20) X (1 − X ) By means of the transformation of Eq. (5) we can obtain random numbers uniformly distributed on the interval (0; 1). Note that this is not a trivial pseudorandom number generator because the numbers Xn were already unpredictable. Fig. 4(b) shows the map (Yn ; Yn+1 ) constructed after transformation (5). Note that the points are uniformly distributed. Moreover, applying the transformation (5) to the sequence generated with the scheme of changing parameter , we obtain uniform random numbers in the same way as with the original time series. We have performed several statistical tests with the functions (9) (after the transformation (5)). Among them are the following: the Central Limit Theorem test, the moments calculation, the variance calculation and the 2 test. The sequence Yn has passed all these tests satisfactorily. With the help of these tests we can verify that {Yn } is a set of mutually independent variables. Now, let us introduce the auto-correlation function: C(m) = hXi Xi+m i − hXi i2 ;
(21)
where h i is an average over all i with i = 1; 2; : : : ; N [24]. The function C(m) measures the correlation between the numbers placed on two sites that are separated by m steps. First, we wish to apply this function to the chaotic map (4) for z = 1:7 (see Fig. 5). In general, for the known chaotic maps, |C(m)| decays with m. Note that there is a range of this dependence that is related to the correlation or memory time. As a rule, if we increase the Lyapunov exponent, the correlation time tends to decrease. However, if we calculate the auto-correlation function for the sequence Yn (using Eqs. (9) and (5)) with z = , then we obtain that the correlation time is exactly zero. Already for m = 1 there are no correlations. In Refs. [25 –27] a new method is developed, which allows to compare the randomness of di erent sequences of equal length. In these works, a measure of randomness (we will call it R) is introduced.
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√ Fig. 5. Autocorrelation function for the map Xn+1 = sin2 (z arcsin X n ) with z = 1:7.
Suppose we have a sequence of values U1 ; U2 ; U3 ; : : : ; UN . Form a sequence of vectors X (i) = [Ui ; Ui+1 ; : : : ; Ui+m−1 ] :
(22)
Now, we will de ne some variables: Cim (r) = (number of j such that d[X (i); X (j)]6r)=(N − m + 1) ;
(23)
where d[X (i); X (j)] is the distance between two vectors, which is de ned as follows: d[X (i); X (j)] = max(|Ui+k−1 − Uj+k−1 |);
(k = 1; 2; : : : ; m) :
(24)
Another important quantity is m (r) =
N −m+1 X
ln Cim (r)=(N − m + 1) :
(25)
i=1
From m (r), we de ne R(m; r; N ) = m (r) − m+1 (r) :
(26)
This measure depends on the resolution parameter r and an “embedding” parameter m. The calculation of R allows to address the system randomness. This technique has been proved to be very e ective in determining system complexity [25 –27]. For given r and m we have a maximum possible randomness. A sequence with maximum randomness is uncorrelated. The randomness generated by our function (9) with z irrational is the maximum possible for the given r and m. The same is obtained for the sequence generated with the scheme of changing parameter . For instance, if r = 0:025, the maximum possible randomness is R = ln 40. The randomness of function (9) with z = approaches the value R = 3:688 for increasing N . For comparison, the
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randomness of the logistic map at the point of full chaos is R = 0:693. Even if we further decrease r and increase m and N , for the logistic map (and other usually chaotic maps), R saturates and remains constant. On the other hand, for r → 0, the randomness of function (9) with z = tends to the maximum possible value, i.e. R → ln(1=r). In the same framework, we can de ne the correlation dimension [28]: ln C m (r) ; (27) r→0 N →∞ ln r PN −m+1 m Ci (r)=(N − m + 1). Here m plays the role of embedding where C m (r) = i=1 dimension. For completely random systems the correlation dimension is equal to the embedding dimension. When we calculate m for function (9) with z = , we nd the relation m = m. In comparison, for the known chaotic systems: m = lim lim
lim m = ;
m→∞
where is a nite number. For suciently large m; m tends to , where .m. Now, we resort to a very important method developed in the papers [29 –31], in order to investigate random and chaotic systems. This technique is very powerful in distinguishing chaos from random time series. The idea of the method is the following. One can make short-term predictions that are based on a library of past patterns in a time series (the method of nonlinear forecasting is described in Refs. [29 –31] and the references quoted therein). By comparing the predicted and actual values, one can make distinctions between random sequences and deterministic chaos. For chaotic (but correlated) time series the accuracy of the nonlinear forecast falls o with increasing prediction–time interval. On the other hand, for truly random sequences, the forecasting accuracy is independent of the prediction interval. If the sequence values are correlated, then future values may, to some extent, be predicted from the behavior of past values that are similar to those of the present. For uncorrelated random sequences the error remains constant. The prediction accuracy is measured by the coecient of correlation between predicted and observed values. For deterministic chaotic sequences this coecient falls as predictions extend further into the future. Suppose we have a sequence U1 ; U2 ; : : : ; UN . Now, we construct a map with the dependence of Un (predicted) as a “function” of Un (observed). If we have a correlated chaotic sequence, this dependence is almost a straight line, i.e. Un (predicted) ≈ Un (observed) (when the forecasting method is applied for one time step into the future). When we increase the number of time steps into the future, this relation deteriorates. The decrease with time of the correlation coecient between predicted and actual values has been used to calculate the largest positive Lyapunov exponent of a time series [30].
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Fig. 6. Predicted values (YnP ) one time step into the future versus observed values (YnO ) for function (9) after the transformation (5) (with z = ).
We have applied this method of investigation to our function (9) after the transformation (5), with z = . Fig. 6shows the predicted values one time step into the future versus observed values. The same picture is obtained for any number m of time steps into the future. As expected, the correlation coecient is independent of the prediction time. Even when the method is applied with a prediction–time interval m = 1, the correlation coecient is zero. This shows that the corresponding time series behaves as a random sequence. We should say that the pseudorandom number generators described in Ref. [10] can pass some of the statistical tests devised to check pseudorandomness [9,11]. However, hidden errors in these generators have been found [10]. Ferrenberg et al. [10] traced the errors to the dependence in the random numbers. Indeed, they are all based on recursive algorithms. We believe that the strongest result of the present paper is the theoretical proof (discussed between Eqs. (11) and (17)) that the outcomes of function (9) are independent. 3. Applications 3.1. Stochastic processes There exists the belief that chaotic processes can be de ned by algorithms, but random processes cannot. On the other hand, it is important to construct pseudo-random number generators and stochastic processes from chaotic systems.
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Chaotic maps have been used as pseudorandom number generators for obtaining samples of variable with di erent distributions [5,11]. Many known random number generators are based on chaotic maps as the following Xn+1 = (aXn + b) mod(T ) :
(28)
Sometimes the very logistic map is used with this aim [5,11,15]. Behind this e ort is the concept that “pseudorandomness is chaos with a very large Lyapunov exponent”. How could actual randomness be de ned. Suppose we have an ideal generator for truly random numbers. In this case no matter how many numbers we have generated, the value of the next number will be still unknown. That is, there is no way to write down a formula that will give the value of the next number in terms of the previous numbers, no matter how many numbers we already have. No recursive algorithm exists that will generate the sequence. The paradigm of random process is the coin toss. But this is not a mathematical procedure that would enable us to make investigations. A chaotic dynamical system as the map (11) can generate a sequence that resemble one generated by an ideal random process. But, given the same initial condition we get the same sequence, unlike a real random process. It is very important to have a process that could be de ned as a standard random process. We think that a practical way out is to use the chaotic functions we are investigating in this paper. We will introduce the concept of a-random sequences which will be free of the limitations of the philosophically inspired de nitions reviewed in Refs. [5,11]. Let us de ne Xn as an a-random sequence if it cannot be a solution of a map of type (11). Hence, there is no recursive algorithm for the generation of an a-random sequence. The ideal random coin toss cannot be expressed as a map of type (11), so it is an a-random sequence. Do we know explicit mathematical functions that represent a-random sequences? The answer is yes! Function (9) for z ¿ 1 fractionary is an a-random sequence. We can use the chaotic function of (9) to model stochastic processes on a computer. 3.2. Monte Carlo method The specialists in Monte Carlo method [9,11] face the problem of testing pseudorandom number generators. How to create a sequence of numbers indistinguishable from those produced by a truly random process. This question possesses practical importance because a very large number of calculations are performed using the Monte Carlo method with pseudorandom numbers. Almost all known generators (in some cases) give rise to incorrect results because they deviate from randomness [10]. There is a lack of mathematical basis for generation of randomness [9,11]. Most of the algorithms used in Monte Carlo calculations are quite predictable. There is also the quasi-Monte Carlo method [11]. The advantage of this method is that the generated points cover the phase space very uniformly and the numerical
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integration has very fast convergence rates. However this algorithm is unappropiate for many Monte Carlo calculations because the generated points are very regular and predictable. The chaotic functions studied here present the required randomness and the generated points cover the space uniformly. 3.3. Cryptography It can be shown [5] that many of the widely used cryptosystems [32,33] are equivalent to chaotic maps. These maps are used to generate a sequence of numbers to be utilized as codes for letters. It does not matter how chaotic a nonlinear map can be, it is not truly random. A text encoded using function (9) cannot be decoded even if we unveil the procedure (but remains unknown). If you do not know it, you know nothing. On the other hand, it is easy to compress and transmit the encoded information. 3.4. Theoretical calculations Functions (9) have also another important application. They are relevant by themselves as theoretical paradigms of stochastic processes. Considering the fact that these are explicit functions, we can use them to solve (analytically) many theoretical problems in stochastic dynamical systems. 4. Discussion and conclusions In the present paper we have studied the chaotic functions Xn = sin2 (z n ) (Eq. (9)). For integer z, these functions represent the general solutions to chaotic maps. The Lyapunov exponent of these maps can be calculated exactly. However, when z is fractionary the produced rst-return map is multivalued. Moreover, if z is irrational the produced rst-return map is an erratic set of points very similar to that produced by noisy systems. Actually, for a fractionary z, it can be shown that function (9) describes a complex dynamics which is in principle unpredictable using the previous values (in this sense, these sequences are di erent from those appearing in the known chaotic systems, where the “unpredictability” is the result of the sensitive dependence on initial conditions). The sequences produced by these functions cannot be generated by a nite recursive algorithm. They are deterministic only in the sense that they are generated by an explicit function. However, there does not exist a map of type (11) to which they are solutions. Therefore, for any nite number r of previous values (Xn ; Xn−1 ; : : : ; Xn−r+1 ), it is impossible to de ne the value Xn+1 as a function of the former. Hence, given r initial values of Xn , the next value is undecidable. There exist in nite values of (in function (9)) that would give the same r initial values of Xn , which, nevertheless, produce
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di erently proceeding sequences. There is no nite method to determine the particular that produced that particular sequence. Thus, there is no method to determine the next value. Several authors [5,11,19] have discussed the di erences between a chaotic sequence and a truly random sequence. The question about the existence of nonrecursive algorithm is fundamental. Here we introduced the concept of a-random sequences to de ne those sequences that can be produced by a determinist algorithm, but not by a recursive algorithm. In fact, function (9), with z ¿ 1 fractionary, satisfy all the known tests for randomness [9]. We have discussed several applications of the obtained results which include: stochastic processes, random number generators, Monte Carlo method, cryptography and theoretical calculations (of course this list could be widely extended). Finally, having in mind that there is no rigorous operational de nition of randomness in terms of classical mathematical primitives [5,11,19], we believe that the introduction of the dynamics generated by function (9) as elementary random process will be very useful and it can serve as a standard for the study of other complex dynamics. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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