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Chaos, Solitons and Fractals 21 (2004) 603–622 www.elsevier.com/locate/chaos

Chaotic and stochastic phenomena in systems with non-invertible non-linearities n J.J. Su arez a, I. Rondo

a,b

, L. Trujillo c, J.A. Gonz alez

a,d,*

a

c

Laboratorio de Fisica Computacional, Centro de Fısica, Instituto Venezolano de Investigaciones Cientıﬁcas, Apartado Postal 21827, Caracas 1020-A, Venezuela b Departamento de Fısica-FACYT, Universidad de Carabobo, Valencia, Codigo Postal 2001, Venezuela P.M.M.H., UMR-CNRS 7636, Ecole Superieure de Physique et Chimie Industrielles, 10 rue Vauquelin 75231, Paris Cedex 05, France d The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34100 Trieste, Italy Accepted 4 December 2003

Abstract We show that systems with non-invertible non-linearities can produce sequences of deterministically independent values. We present autonomous dynamical systems that exhibit random behavior in such a way that all variables (taken separately) are unpredictable. We study a new mechanism for chaos in which a static system without inertial or dynamical elements can produce complexity. Using some results of exactly solvable chaotic systems we investigate the inﬂuence of diﬀerent chaotic signals on the phenomenon of stochastic resonance. We report the results of real experiments concerning all these phenomena. Ó 2003 Elsevier Ltd. All rights reserved.

1. Introduction Few abstract concepts from frontier scientiﬁc research have produced such a series of important practical results as deterministic chaos [1–19]. The chaos that arises in the context of non-linear systems represent sometimes such a complicated motion that seems beyond human comprehension. Is this motion random? The present paper is dedicated to the relationship between chaos and randomness and to the production of complexity from simple initial conditions and inputs. There exist real physical systems in which the knowledge of an arbitrary large number of previous outcomes is still insuﬃcient to predict the further behavior of the system. Can this kind of behavior be reproduced in non-stochastic dynamical systems? We are interested in the emergence of random behavior in simple systems without external noise or stochastic ﬂuctuations. Does determinism imply predictability? The known chaotic systems are predictable at least in the short term. The common believe is that for modelling diﬀerent random processes or signals, diﬀerent dynamical systems are needed, that is, systems with diﬀerent number of degrees of freedom. White noise poses the hardest problem for

* Corresponding author. Address: Laboratorio de Fisica Computacional, Centro de Fısica, Instituto Venezolano de Investigaciones Cientıﬁcas, Apartado Postal 21827, Caracas 1020-A, Venezuela. Fax: +58-212-5041148. E-mail address: [email protected] (J.A. Gonzalez).

0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.006

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dynamical reproduction since it would require an inﬁnite number of degrees of freedom implying that its dimension has to be inﬁnite. Many real systems can behave in an irregular fashion with all the signatures of truly random behavior. Where does this randomness come from? Truly random time-series are diﬀerent from the chaotic ones. If we have a sequence of the past values X0 ; X1 ; X2 ; . . . ; Xm ; even the next value Xmþ1 cannot be determined from the knowledge of the previous values. It does not matter how many values are already known [20–23]. In the present paper we will show that very simple autonomous systems are able to exhibit random behavior in such a way that all variables (taken separately) are unpredictable. This randomness is an intrinsic property of the systems and is not imposed on them by stochastic surrounding or through external noise. We will also study a new scenario for chaos in which a static system without inertial or dynamical elements can produce complexity. In these systems a regular signal is used as the input to a non-linear system that will transform the signal in such a way that the output can be written analytically as an explicit function of the input. These systems can be realized experimentally using non-invertible I–V characteristics. Using chaotic systems for which we calculate exactly the Lyapunov exponent, we have been able to investigate the inﬂuence of diﬀerent levels of chaos on the phenomenon of stochastic resonance. We report the result of real experiments.

2. Sequences of deterministically independent values Recently, we have introduced explicit functions that represent truly random sequences [24–28]. We have been investigating the following function Xn ¼ sin2 ðhpZ n Þ;

ð1Þ

where Z is a real number and h is a real parameter. For a Z > 1, this is the solution to some chaotic maps (see for example [29]). We will present here a proof of the fact that the sequences generated by function (1) with noninteger Z are unpredictable from previous values. Let Z be a rational number expressed as Z ¼ pq, where p and q are relative prime numbers. We are going to show that if we have m þ 1 numbers generated by function (1) X0 ; X1 ; X2 ; . . . ; Xm (m can be as large as we wish), then the next value Xmþ1 is still unpredictable. This is valid for any string of m þ 1 numbers generated by function (1): Let us deﬁne the following sequences n p 2 ðk;mÞ m Xn ; ð2Þ ¼ sin pðh0 þ q kÞ q where k is an integer. The parameter k distinguishes the diﬀerent sequences. For all sequences parameterized by k, the ﬁrst m þ 1 values are the same, this is so because n n p p þ pkpn qmn ¼ sin2 ph0 ; Xnðk;mÞ ¼ sin2 ph0 q q

ð3Þ

for all n 6 m. Note that the number kpn qmn is an integer for n 6 m. So we can have an inﬁnite number of sequences with the same ﬁrst m þ 1 values. Nevertheless, the next value # " mþ1 p pmþ1 ðk;mÞ 2 Xmþ1 ¼ sin ph0 ð4Þ þ pk q q is uncertain. ðk;mÞ In general, Xmþ1 can take q diﬀerent values. From the observation of the previous values X0 ; X1 ; X2 ; . . . ; Xm , there is no method for determining the next value. If Z is irrational, we can have an inﬁnite number of diﬀerent outcomes (see Fig. 1). Now we wish to show that these values are independent. If we generated a simple chaotic sequence Y0 ; Y1 ; Y2 ; . . . ; Yn with the logistic map: Ynþ1 ¼ 4Yn ð1 Yn Þ, it is not random. However, if now we create a new sequence X0 ¼ Yr ; X1 ¼ Yr1 ; X2 ¼ Yr2 ; . . . ; Xm ¼ Y0 ; then the sequence Xn cannot be predicted. After any value of Xm , we have two possible values for Xmþ1 . Nevertheless, this is hardly a random sequence.

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Fig. 1. First-return map produced by function (1) with an irrational Z.

The values are not independent. In this case, the previous values are determined by the futures values. We wish to show that the function (1) represents not only unpredictable sequences, but also sequences of independent values. In this paper we will call a sequence deterministically if for any string Xs ; Xsþ1 ; Xsþ2 ; . . . ; Xsþm if the next value Xsþmþ1 cannot be determined by the previous values, and the value Xs1 also cannot be determined by the future values. In the same spirit of Eq. (2), let us deﬁne the following family of sequences s n q p 2 k;m;s m ; ð5Þ ¼ sin pðh0 þ q kÞ Xn p q where k, m and s are integer. For all sequences parameterized by k, the string of m þ 1 values Xs ; Xsþ1 ; Xsþ2 ; . . . ; Xsþm are the same. Nevertheless, the next value Xsþmþ1 cannot be determined. There can be q diﬀerent possible values. In addition, the value Xs1 is also unpredictable from the values of the string Xs ; Xsþ1 ; Xsþ2 ; . . . ; Xsþm . In fact, there are p diﬀerent possible values for Xs1 . The future and the past are both uncertain. In the case of an irrational Z, there are inﬁnite possibilities for the future and the past. The future values and the past values are independent of this string of values. And this is valid for any string. And m can be any number. In other words, this result shows that for any given string of values, there exist always an inﬁnite number of values of h for which function (1) can produce the same string. The rest of the time-series (past values and the future values) for diﬀerent h is diﬀerent in most of the cases. This property is in part related to the fact that equation sin2 h ¼ a, where 0 6 a 6 1 possesses inﬁnite solutions for h. The sequence produced by function (1) is random but pis uniformly distributed. The probability density behaves ﬃﬃﬃﬃﬃnot 1 ﬃ. After transformation Yn ¼ p2 arcsin Xn , we obtain P ðY Þ ¼ 1 (see Fig. 2). as P ðX Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ð1X Þ

Fig. 2. First-return map constructed with the sequence Yn ¼ p2 arcsin irrational Z.

pﬃﬃﬃﬃﬃ Xn , where the sequence Xn is produced by function (1) with an

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The argument of function (1) does not need to be exponential all the time for n ! 1. In fact, a set of ﬁnite sequences (where each element-sequence is unpredictable, and the law for producing a new element-sequence cannot be obtained from the observations) can form an inﬁnite unpredictable sequence. So using (1) we can produce very long random sequences changing the value of h after a ﬁnite number N of values of Xn . This procedure can be repeated the desired number of times. It is important to have a non-periodic way for generating the new values of h. For example, we can use the following method in order to change the parameter h after each set of N sequences values. Let us deﬁne hs ¼ AWs , where Ws is produced by a random procedure and s is the order number of h in the way that s ¼ 1 corresponds to the h used for the ﬁrst set of N values of Xn , s ¼ 2 for the second set, etc. For the numbers Ws we can use the same random values produced with function (1) i.e. we can put W1 ¼ X1 , W2 ¼ X2 , etc. The inequality A > 1 should hold to ensure the absolute unpredictability. In this case, from the observation of the values Xn , it is impossible to determine the actual value of h.

3. The role of non-invertible non-linearities What are the properties of function (1) that lead to the random behavior? Let us write the function Xn in a very general form Xn ¼ hðf ðnÞÞ:

ð6Þ

A deep analysis of function (6), in the light of the properties of function (1), shows that function f ðnÞ does not have to be exponential all the time, and function hðyÞ does not have to be periodic. In fact, it is suﬃcient for function f ðnÞ to be a ﬁnite non-periodic oscillating function which possesses repeating intervals with ﬁnite exponential behavior. For instance, some chaotic functions have these properties. On the other hand, function hðyÞ should have several maxima and minima in such a way that equation hðyÞ ¼ a (for some speciﬁc interval of a, (a1 < a < a2 ) possesses several real solutions for y. The following autonomous dynamical system can produce truly random dynamics: aXn ; if Xn < Q; Xnþ1 ¼ ð7Þ bYn ; if Xn > Q; Ynþ1 ¼ cZn ;

ð8Þ

2

ð9Þ

Znþ1 ¼ sin ðpXn Þ:

Here a > 1 can be an irrational number, b > 1, c > 1. We can note that for 0 < Xn < Q the behavior of function Zn is exactly like that of function (1). For Xn > Q the dynamics is re-injected to the region 0 < Xn < Q. The dynamics of Zn is random as is function (1). On the other hand the process of producing a new initial condition through the Eq. (8) is also random. If the only observable is Zn , then it is impossible to predict the next values of this sequence using only the knowledge of the past values. An example of the dynamical system (7)–(9) is shown in Fig. 3. Moreover, we also can construct a dynamical system where all the variables (taken separately) are random: ða þ bZn ÞXn þ cYn ; if Xn < Q; ð10Þ Xnþ1 ¼ bYn ; if Xn > Q; Ynþ1 ¼ cZn ;

ð11Þ

Znþ1 ¼ sin2 ðpXn Þ:

ð12Þ

Note that Xn , in Eq. (10), still possesses a ﬁnite exponential behavior for 0 < Xn < Q, because a þ bZn is always a positive number. However, in this case the dynamics of Xn is inﬂuenced all the time by random dynamics of Zn . Firstreturn maps of the times-series produced by dynamical system (10)–(12) can be observed in Fig. 4. If we are interested in dynamical systems, where all the variables are random and uniformly distributed, then we can use the following one: Xnþ1 ¼ ½ða þ bZn ÞXn þ cYn þ 0:1ðmod 1Þ;

ð13Þ

Ynþ1 ¼ ½dZn þ 0:1ðmod 1Þ;

ð14Þ

Znþ1 ¼ ½fXn þ 0:1ðmod 1Þ:

ð15Þ

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Fig. 3. First-return maps produced by the sequence Zn in dynamical system (7)–(9). (a) a ¼ 4=3, b ¼ 2, c ¼ 2, Q ¼ 100. (b) a ¼ p, b ¼ 2, c ¼ 3, Q ¼ 100.

Here Xn shares many of the properties that are present in the system (10)–(12). Function Y ¼ X ðmod 1Þ used in Eq. (15) is periodic and non-invertible as required. First-return maps of the times-series produced by dynamical system (13)–(15) can be observed in Fig. 5.

4. Physical systems Can we ﬁnd a physical system where there is a natural sine-function? The fundamental equations for the Josephson junctions [30] are the following: I ¼ Ic sinð/Þ;

ð16Þ

d/ 2eV ¼ ; dt h

ð17Þ

where I is the current in the junction, Ic is the critical current, V is the voltage across the junction, / is the phase diﬀerence of the superconducting order parameter between each side of the barrier, e is the charge of the electron and h is the Planck’s constant. In order to have the argument of sine-function in Eq. (16) behaving as a chaotic time-series, we can apply a chaotic voltage V ðtÞ across the junction. In the last decades there have been a wealth of experimental work dedicated to the creation of electronic analog that can simulate the Josephson junction [31–34]. In that case the Eq. (17) is replaced by equation d/ ¼ kV , which is related dt to certain integrator time constant RC in the circuit. We have performed real experiments with a non-linear chaotic circuit coupled to an analog Josephson junction. In our experiments we have used the Josephson junction analog constructed by Magerlein [34]. This is a very accurate device that has been found very useful in many experiments for studying the junction behavior in diﬀerent circuits. The

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Fig. 4. First-return maps produced by the sequence Zn in dynamical system (10)–(12). (a) a ¼ 7=3, b ¼ 111, c ¼ 70, Q ¼ 500, (b) a ¼ p, b ¼ 111, c ¼ 70, Q ¼ 500.

junction voltage is integrated using an appropriate resetting circuitry to calculate the phase /, and a current proportional to sin / can be generated. The circuit diagram can be found in [34]. We need large values of k in order to increase the eﬀective domain of the sine-function. In other words, we need the argument of the sine-function to take large values in a truncated exponential fashion. This allows us to have a very unpredictable output signal. In our case, k ¼ 104 . The voltage V ðtÞ will be produced by a chaotic system. In our case we selected the Chua’s circuit [35]. We implemented the Chua’s circuit using the recipe of Ref. [36]. The following components were used: C1 ¼ 10 nF, C2 ¼ 100 nF, L ¼ 19 mH and R is a 2.0 kX trimpot with R 1800 X. In our experiment the voltage in C1 was used as a driving signal for the Josephson junction. The results of the experiment are shown in Fig. 6 which is the ﬁrst return map of the time-series produced by direct measurements of the junction current. The time-interval between measurements was 10 ms. If we apply the non-linear forecasting method analysis [20–23] to a common chaotic system, then the prediction error increases with the number of time-step into the future. On the other hand, when we apply this method to the time-series produced by our system, the prediction error is independent of the time-steps into the future, as in the case of a random time-series. Other very strong methods [37,38], which allows to distinguish between chaos and random noise, yield the same result. In conclusion, we have proved that function (1) represents a random dynamics where any string of values Xs ; Xsþ1 ; Xsþ2 ; . . . ; Xsþm of any length m þ 1 is independent of any other string values in the past or in the future. Dynamical systems consisting of chaotic subsystems coupled to a subsystem with non-invertible non-linearities can produce similar dynamics. We can construct dynamical systems where all the variables (taken separately) are random. We have shown that a Josephson junction coupled to a chaotic system can produce completely random dynamics. These results possess many practical applications e.g. in secure communications [39–41] dynamics-based computation [42–45]. In fact we have performed many experiments with several physical systems.

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Fig. 5. Dynamics produced by the dynamical system (13)–(15). All variables are unpredictable.

For our experiments we only need a source of chaotic dynamics which contains repeating intervals of ﬁnite exponential behavior and a physical system able to make a transformation of the chaotic signal equivalent to the application of a non-invertible function. The scheme of this system is shown in Fig. 7. If we use the scheme of Fig. 8, then the dynamical variables of the chaotic system will become random too. This can be seen in the dynamical systems discussed in the paper. An example of this structure can be the non-linear system constructed with non-linear circuits shown in Fig. 9. The chaotic circuit of the left can be one of those described in Ref. [46]. These chaotic circuits produce chaotic maps equivalent to many well-known non-linear maps: The logistic map, the cubic map, etc. The circuit of the right should possess a non-invertible I–V characteristic as in Fig. 10.

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Fig. 6. First-return map of the Josephson junction output current.

Fig. 7. Experimental setup to produce truly random dynamics.

Fig. 8. Experimental setup to produce truly random dynamics. All the variables are random.

Fig. 9. Non-linear system using non-linear circuits.

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Fig. 10. Non-linear non-invertible I–V characteristics.

Fig. 11. Examples of non-linear circuits that possess non-invertible I–V characteristics.

In Ref. [47] we can ﬁnd diﬀerent methods to construct circuits with these I–V characteristics. For example, in Fig. 11 we can see circuits with non-invertible characteristics with two extrema. The circuit shown in Fig. 12 is equivalent to a transformation with several extrema. There are electronic systems designed to produce almost any elementary function (Refs. [48–60]). This includes the sine-function. So we can mimic the functions: Xn ¼ sin2 ðhpZ n Þ:

ð18Þ

Using these systems we can obtain dynamical behaviors similar to that generated by function (18) and dynamical systems (Eqs. (7)–(15)). We have constructed several circuits with non-invertible I–V characteristics including the ones shown in Figs. 11 and 12. We produced chaotic time-series using a common non-linear map and then we transformed them into analog signals using a converter. The same signals can be produced by the circuits described in Ref. [46]. These analog signals were introduced as the voltage-input to the circuit shown in Fig. 11. We have a model for this experiment:

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Fig. 12. Circuit that possesses a multiple-maxima–minima I–V characteristic.

Xnþ1 ¼ aXn ½1 hðXn qÞ þ bYn hðXn qÞ; h pﬃﬃﬃﬃ i Ynþ1 ¼ sin2 d arcsin Yn ;

ð19Þ

Znþ1 ¼ 4Wn3 3Wn ;

ð21Þ

ð20Þ

where Wn ¼ 2Xn =s 1, q ¼ s=a, s ¼ 10, b ¼ 7, a ¼ p=2, d ¼ 3, hðkÞ is the Heaviside function. Fig. 13 shows the comparison between the modelling, and the experiment. This dynamics is unpredictable but not ‘‘inﬁnitely’’ unpredictable in the sense that in this case there is some ‘‘pattern’’ in the return-map. However this allows us to check the fact that the experiment is producing exactly what we expect. We can produce ﬁrst-return maps as that shown in Fig. 6 using electronic systems that simulate the sine-function or using a Josephson junction. It is well-known that the current in a Josephson junction may be calculated as: I ¼ Ic sinð/Þ;

ð22Þ

¼ kV , / is the phase and V is the voltage across the junction. where d/ dt It is always possible to use a non-linear circuit such that the transformation undergone by the signal in the system of the right in Fig. 9 is the desired one. For instance, we can produce time-series equivalent to the following functions: Xn ¼ sin2 ðhpZ n Þ;

Fig. 13. (a) Dynamics produced by system (19)–(21), (b) dynamics produced by the system in the circuit (see Fig. 11a).

ð23Þ

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Xn ¼ ln½tan2 ðhpZ n Þ;

613

ð24Þ

n

Xn ¼ tanðhpZ Þ

ð25Þ

and Xn ¼

hpﬃﬃﬃﬃ i 2 arcsin Y n ; p

ð26Þ

where Yn ¼ sin2 ðhpZ n Þ. We should stress that all these functions are realizable (Refs. [48–60]) in non-linear circuits and the experiments with them are quite clean. In fact, these circuits are designed for analog computing.

Xn+1

1

0.5

0

0

0.5

Xn Fig. 14. First-return map of the stochastic function (23).

Fig. 15. First-return map of the stochastic function (24).

1

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It is very important to note that our theory says that the random output sequences in all these cases possess very 1 ﬃ. diﬀerent distributions. For instance, the probability density P ðX Þ for the sequence (23) behaves as P ðX Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ð1X Þ

In the case (24), the distribution is a near Gaussian function. Function (25) produces values with a distribution which is a power-law P ðX Þ X1a . On the other hand, the values produced by function (26) possess the probability density P ðX Þ ¼ 1. So the values are uniformly distributed. This can be observed in Figs. 14–17. Note that in Fig. 14 the dots are more dense around the values X ¼ 0 and X ¼ 1. All the experiments agree with these distributions.

Fig. 16. First-return map of the stochastic function (25).

Yn+1

1

0.5

0

0

0.5

Yn Fig. 17. First-return map of the stochastic function (26).

1

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615

5. A new mechanism for producing chaos Another consequence of our theory is that a quasi-periodic time-series can be transformed into a chaotic and very complex dynamics by non-linear circuits with non-invertible characteristics. The theoretical result is that the following function is chaotic: X ðtÞ ¼ P ðA exp½QðtÞÞ;

ð27Þ

where P ðyÞ is a periodic function (in some cases it can be just a non-invertible function), and QðtÞ is a quasi-periodic function. Let us introduce here some explanation of this result. We have seen that the exact solutions to several chaotic maps can be written using the following functions: Xn ¼ sin2 ðhpZ n Þ;

ð28Þ

where n is the discrete time. For some Z these functions are more than chaotic, they are random. Can we write down chaotic functions with a continuous time t? The obvious observation leads us to think that we need an exponential argument in the sine-function. The trivial example is the following: X ðtÞ ¼ sinðh expðbtÞÞ:

ð29Þ

However, it is hard to believe that a continuous real physical process can be described by this function, because the frequency of the oscillations of variable X ðtÞ tends to inﬁnite as t ! 1. We do not know any physical phenomenon with this property. So we need a function that behaves exponentially for a ﬁnite interval of time. On the other hand, if we are looking for a continuous chaotic function that represents a sustained oscillation (for all t), then the argument of the sinefunction should possesses repeating intervals with exponential behavior. For instance, the following functions possess repeating intervals of exponential behaviors. UðtÞ ¼ A expðB½sinðxtÞÞ:

ð30Þ

In fact, functions sinðxtÞ behaves approximately as an increasing linear function whenever xt 2pk where k is an integer. However, functions (30) are periodic. And so the function X ðtÞ ¼ sin½UðtÞ will be also periodic. This function will have intervals that seem chaotic. But, these seemly chaotic regimes will repeat periodically. Thus at the end, the whole function will have a period. So we need an oscillating function QðtÞ which is not periodic. In this case, the function: UðtÞ ¼ A exp½QðtÞ

ð31Þ

will have non-periodically repeating intervals of exponential behaviors. The following function possesses these properties QðtÞ ¼ a1 sinðx1 tÞ þ a2 sinðx2 tÞ þ a3 sinðx2 tÞ, where the frequencies x1 , x2 and x3 are such that their ratios xx21 , xx32 , xx31 are irrational numbers. In this case, the function X ðtÞ ¼ sinðA exp½QðtÞÞ will be a (non-periodic) chaotic process. This result can be generalized to other class of functions. We can have very general chaotic functions as the following: X ðtÞ ¼ P ½UðtÞ;

ð32Þ

where UðtÞ ¼ A exp½QðtÞ, QðtÞ is a non-periodic oscillating function and P ðyÞ is a periodic function. We have constructed a chaotic function that is a particular case of (32) and with the feature that we can calculate its Lyapunov exponent analytically. We will study the following function: X ðtÞ ¼ UðtÞðmod 1Þ;

ð33Þ

UðtÞ ¼ A exp½QðtÞ

ð34Þ

where and QðtÞ ¼ P1 ðtÞ þ P2 ðtÞ þ P3 ðtÞ, Pi ðtÞ ¼ at akTi when kTi 6 t 6 ðk þ 1ÞTi , and k is an integer number. The ratios TT21 , and TT31 are irrational numbers.

T3 T2

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This functions are represented in Figs. 18–21. The theoretical Lyapunov exponent of the dynamics produced by function (33) is given by the formula: k ¼ ln 3a:

ð35Þ pﬃﬃﬃ We generated a numerical time-series using function (33) with a ¼ 1, T1 ¼ p, T2 ¼ 3 þ 2, T3 ¼ 3. Then we calculated the Lyapunov exponent of the generated time-series using standard numerical methods. These calculations corroborate the theoretical result that k ¼ ln 3. We have performed the experiments using the scheme of the Fig. 22. In our experiment a quasi-periodic time-series was used as the input to an electronic circuit that simulates an exponential function. The output of the exponential system is taken as the input to a non-linear system that simulates the sine-function. The circuit with the exponential transformation used in the experiment is the one shown in Fig. 23. We used several quasi-periodic signals including piece-wise linear functions which allow us to make important theoretical calculations. However the most ‘‘natural’’ example of function (27) is the following: X ðtÞ ¼ sinðA exp½a1 sinðx1 tÞ þ a2 sinðx2 tÞ þ a3 sinðx3 tÞÞ:

ð36Þ

The function QðtÞ ¼ a1 sinðx1 tÞ þ a2 sinðx2 tÞ þ a3 sinðx3 tÞ is the input to the ‘‘exponential circuit’’.

4 P1(t)

3 2 1 0 0 4

10

20

30

10

20

30

20

30

P2(t)

3 2 1 0 0 4 P3(t)

3 2 1 0

0

10 t

Fig. 18. Periodical signals used to produce a quasi-periodic signal for the input of the system of Fig. 22. See Eq. (34).

10

Q(t)

8

6

4

2

0

0

20

40

60

80

100

t

Fig. 19. Quasi-periodic signal used as input of the system shown in Fig. 22. See Eq. (34).

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11000 10000 9000 8000

Φ

7000 6000 5000 4000 3000 2000 1000 0

0

20

40

60

80

100

t

Fig. 20. Signal produced by the function /ðtÞ in Eq. (34).

1

X(t)

0.8

0.6

0.4

0.2

0

0

10

20

30

40

50

t Fig. 21. Chaotic signal produced by function (33).

Fig. 22. Scheme of the experimental setup to produce chaos using quasi-periodic signals and non-linear static systems.

The fact that we can produce chaos from quasi-periodicity using some non-linear ‘‘static’’ systems without inertial elements is by itself a very important result. However, in the present discussion, the relevant fact is that we can calculate theoretically the Lyapunov exponent of the output chaotic signal and we can compare it with the Lyapunov exponent of the experimental time-series. Moreover, in the chaotic time-series that we can create with function (36) there are patterns and other dynamical features. These patterns and dynamical features can be observed in the experimental time-series generated by the physical system represented in Fig. 22. See an example of the chaotic time-series generated in our experiments in Fig. 24.

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Fig. 23. Exponential-converter circuit diagram.

1

I (mA)

0.5

0

–0.5

–1

0

500

1000 t (ms)

1500

2000

2500

Fig. 24. Experimental chaotic time-series produced by the output of the system represented in Fig. 22.

6. Stochastic resonance and chaos Function Xn ¼ sin2 ðhpZ n Þ, with a non-integer Z, is able to produce random dynamics. However, for integer Z the dynamics is just a common chaotic (but deterministic) time-series. By ‘‘deterministic’’ here we mean that the previous values determine the future pﬃﬃﬃﬃ values. Besides, our theory leads to the construction of exactly solvable chaotic maps of type Xnþ1 ¼ sin2 Z arcsin X n for which it is possible to exactly calculate the Lyapunov exponent: k ¼ ln Z. Here Z can be any number Z P 1. Using all this framework we have been able to investigate analytically, stochastic resonance systems. In stochastic resonance systems, the sum of noise (or chaos) and a weak periodic signal is used to drive a non-linear system (see Fig. 25). In early papers, this non-linear system used to be a bistable system. The most important characteristic of stochastic resonance is that the signal-to-noise ratio (SNR) has a maximum in the plot SNR vs D, where D, is the noise intensity, for a ﬁnite non-zero value of the noise intensity. Now we know that stochastic resonance can appear not only in bistable systems. Stochastic resonance is possible also in so-called threshold systems [61–63]. For instance the non-linear system that appears in Fig. 25 can be a circuit shown in Fig. 26, which is a concave resistor. The I–V characteristic of this circuit can be described by the following equation: 0; for V < Vth ; y¼ ð37Þ bðV Vth Þ; for V > Vth : This I–V characteristic can be seen in Fig. 27. A very important question is how stochastic resonance depends on the Lyapunov exponent of the driving chaotic noise.

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Fig. 25. Scheme of the experimental setup for stochastic resonance.

Fig. 26. Concave diode.

Fig. 27. I–V characteristic of the concave diode.

We have investigated the explicit function tanhðBVn Þ þ 1 bVn ; In ¼ 2

ð38Þ

where Vn ¼ A sinðxn Þ þ DXn , Xn is a chaotic signal constructed using our functions and systems with exactly calculable Lyapunov exponents. We have found theoretically that in the interval lnð32Þ 6 k 6 lnð2Þ, the stochastic resonance strongly depends on the Lyapunov exponent. In this interval, the maximum of SNR is ampliﬁed as we decrease the Lyapunov exponent. We have performed experiments with the system shown in Fig. 25 with the the circuit shown in Fig. 26, as the nonlinear system. For the noise shown in the scheme of Fig. 25 we can pﬃﬃﬃﬃ dynamics produced in our experiments that use therandom simulates function Xn ¼ sin2 ½hpZ n , the map Xnþ1 ¼ sin2 Z arcsin X n and the experiments where we have been able to produce chaos from quasi-periodicity.

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J.J. Suarez et al. / Chaos, Solitons and Fractals 21 (2004) 603–622 0.2 Z=1.5 Z=1.7 Z=2.0 Z=2.2

SNR

0.15

0.1

0.05

0

0

1

2

3

4

5

D(Vrms) Fig. 28. SNR vs noise intensity (D) for diﬀerent values of Z.

3

I(mA)

2.5

2

1.5

1

0.5 00

500

1000

1500

2000

t(ms) Fig. 29. Time-series of the stochastic resonance experiment.

In all of these experiments we know when we have chaos, when we have random dynamics and what are the Lyapunov exponents of the experimental time-series. The results of our experiments with the concave resistor are shown in Fig. 28. We can observe clearly the ampliﬁcation of the SNR, when we reduce the Lyapunov exponent in the interval lnð32Þ 6 k 6 lnð2Þ. Fig. 29 shows the patterns that we have observed in our experiments. Note that the most regular time-series is obtained for an intermediate value of the noise intensity.

7. Conclusion The recognition that even very simple systems, describable by fully deterministic physical laws, can exhibit behavior that is unpredictable, random, and of apparently limitless complexity, could represent a true revolution in our view of the world around us. One of the most remarkable progress in the future of non-linear science could be a signiﬁcant advance in the understanding of randomness. A very remarkable question is what are the sources which give rise to unpredictability and randomness? We believe that in the present paper we have described at least one of the possible mechanisms. On the other hand, we have addressed the question: How complexity can be born out of non-complexity.

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We have shown that a static system of type Yn ¼ /ðXn Þ, where Xn is a regular signal (the input), /ðxÞ is a non-linear transformation, can produce a chaotic output which is time-series Yn . We also present some experimental results about the generation of complexity in systems with noinvertible non-linearities and stochastic resonance systems.

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