Laplace's Deterministic Paradise Lost ?

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“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements Laplace’s deterministic paradise lost of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”

Miguel Angel Fuentes Santa Fe Institute, USA Pierre-Simon Laplace, ~1800 Instituto Balseiro and CONICET, Argentina

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Comments: Heisenberg uncertainty principle

In the framework of quantum mechanics -one of the most successful created theories- is not possible to know -at the Einstein was very unhappy about this apparent randomness same time- the position and velocity of a given object. (~1927) in nature. His views were summed up in his famous phrase, 'God does not play dice.'

∆v ∆x m > h v v v (from Stephen Hawking’s lecture)

Santa Fe Institute Santa Fe Institute In chaotic (deterministic) systems

Same functionality (SAME ORGANISM) x(t)

Not chaotic trajectory

x(0) = c ∆x ± t

? Some expected initial condition + Mutations

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Same functionality (SAME ORGANISM)

Many process can happen between regular-chaotic behavior

Chaotic trajectory

Some expected initial condition + Mutations

Santa Fe Institute Santa Fe Institute 4.1 Parameter Dependence of the Iterates 35 4.1 Parameter Dependence of the Iterates 35

r values are densely interwr values oarevedenselyn andinterwoneovfindsen and onea sensitifinds a sensitive dependenceve dependence ononparameterparametervalues.values. We also discuss the conceptWe also discussof structuralthe conceptuniof structuralversalityuniversalityand calculateand calculatethethe invvariantariantdensitydensityof the of the logistic map at r = 4. logistic map at r = 4. Finally, in Section 4.5 we present a summary that explains the parallels between the Finally, in SectionFeigenbaum4.5 weroutepresentto chaosaandsummaryordinary equilibriumthat explainssecond-orderthephaseparallelstransitions.betweenThis the Feigenbaum route to chapterchaosendsandwithordinarya discussionequilibrium4.1of the measurableParametersecond-orderpropertiesDependenceof the Feigenbaumphaseof thetransitions.Iterrouteatesand a This 35 review of some experiments in which this route has been observed. chapter ends with a discussion of the measurable properties of the Feigenbaum route and a review of some experiments in which this route has been observed. r values are densely4.1interwParameteroven andDependenceone finds a ofsensitithe Iteratesve dependence on parameter values.

We also discuss the conceptTo provide anofovstructuralerview in this section,universalitywe present andseveralcalculateresults for thethelogisticinvmapariantobtaineddensity of the logistic map at r = 4.by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the 4.1 ParameteraccumulationDependencepoints of the iteratesoff nthe(x ) forIteratesn > 300 as a function of r together with the Finally, in Section 4.5 we present a summary{ r 0 } that explains the parallels between the Some process are neitherLiapuno chaoticv enorxponent regularλ obtained via eq. (3.9). TFeigenbaumo provide an oroutevervietow chaosin this section,and ordinarywe presentequilibriumseveral resultssecond-orderfor the logisticphase maptransitions.obtainedThis x (t) x (t) eλt bychaptercomputerendsiterationwith a ofdiscussioneq. (4.1) forof thedifferentmeasurablevalues ofpropertiesthe parameterof ther.FeigenbaumFigure 23 shoroutews theand a green blue accumulationreview of somepointsexperimentsof the iteratesin whichf n(xthis) routefor n >has300beenas observa functioned. of r together with the | − | ∼ { r 0 } Liapunov exponent λ obtained via eq. (3.9). λ : Lyapunov exponent, positive for chaos 4.1 Parameter Dependence of the Iterates

To provide an overview in this section, we present several results for the logistic map obtained by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the n x(t) accumulation points of the iterates f (x0) for n > 300 as a function of r together with the { r } 4.1 Parameter Dependence of the Iterates 35 Liapunov exponent λ obtained via eq. (3.9).

Santa Fe Institute Santa Fe Institute 4.1 Parameter Dependence of the Iterates Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.).

To provide an overview in this section, we present several results for the logistic map obtained by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the accumulation points of the iterates f n(x ) for n > 300 as a function of r together with the { r 0 } There are importantLiapuno theorv exponentems for randomλ obtained variablesvia eq. that(3.9). are valid for chaotic systems Until now we were discussing:

Chaos and uncertainty

Its connection with the idea of random processes Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Chaos and Lyapunov exponent

I mention that some important theorem for random variables are valid for deterministic variables too. Let move to one of them (VERY important one.) Here

Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.).

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Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Abraham de Moivre (1667) a french mathematician famous for de Moivre's formula (friend of Isaac Newton, Edmund Halley, and James Stirling, nice crowd!) Measurement n (cos x + i sin x) = cos nx + i sin nx x1 x 1.0 2 0.8

0.6 RAPID COMMUNICATIONS xn 0.4 TIRNAKLI, BECK, AND TSALLIS PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒ 0.2 The central limit theorem states that the sum of a large prove the existence of a CLT. This means the probability 6 6 !3 !2 !1 N=2x10 ; nini=12x10 2 3 number of independent and identically-distributed random 6 distribution of N=100 ; nini=10x10 eq.(9) ; N=100 variables will be approximately normally distributed (i.e., N 100 6 following a Gaussian distribution) if the random variables 1 eq.(9) ; N=2x10 y ª ͚ f͑xi͒ ͑2͒ ͱN i=1 Normal distribution have a ﬁnite variance (~1773). a=2 System 10-1 becomes Gaussian for N ϱ, regarding the initial value x1 → d k )

as a random variable. Here f : is a suitable smooth y

R →R (

function with vanishing average which projects from the y d-dimensional phase space to a k-dimensional subspace. If ρ 10-2 Santa Fe Institute d=k=1, the variance ␴2 of this Gaussian is given by ͓8͔ Santa Fe Institute ϱ 2 2 10-3 ␴ = ͗f͑x0͒ ͘ + 2͚ ͗f͑x0͒f͑xi͒͘. ͑3͒ i=1 Here denotes an expectation formed with the natural RAPID COMMUNICATIONS ͗¯͘ 10-4 N invariant density of the map T. x -4 -2 0 2 4 Example: z = 1 i TIRNAKLI,As an BECK,exampleANDtoForillustrateTSALLIS the logisticthese generalmap atresults, positivelet Lusyapunov exponent (CHAOS)PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒ √ consider the logistic map ! N y prove the existence of a CLT. This means2 the probability N=2x106 ; n =2x106 xi+1 = T͑xi͒ = 1 − axi ͑4͒ FIG. 1. ͑Color online͒ Probabilityini density of rescaled sums of distribution of N=100 ; n =10x106 iterates of the logistic*0 mapini with a=2 as given by Eq. ͑8͒, on the interval X=͓−1,1͔. For a=2 the system is ͑semi͒con- 6 eq.(9) ; N=100 N N=2ϫ10 and100 N=100. *The5'67number7*\:(*-\*+,of-.$initial$!!!!values/ contributing to jugated to a Bernoulli shift1and strongly mixing. The natural eq.(9) ; N6=2x106 7 PN (z) the histogram is nini=2ϫ10 , respectively nini=10 . The solid lines y f͑xi͒ ͑2͒ invariant density is ª ͚ correspond to Eq. ͑9͒. &'(')*+,-.$$!!!!/0 ͱN i=1 1-234)*5'677 a=2896'(:-;)*<=*?@A?BC79,(?DEAFGGGC3H+?IF 1 10-1 J3:KL(:;K)* becomes Gaussian for ␳N͑x͒ ϱ=, regarding. the initial value͑5x͒ < M-*B3:KL(:;K x 2 $"% 1 13,14 . Our ﬁnal result** is that for ﬁnite but large N the ͓ ͔ ) → ␲ͱd1 − x k NL:OFA&-P =*!"!!!F

as a random variable. Here f : is a suitable smooth y probability density ofQOFy is=**!given"RRRS by R →R ( ** function with vanishing average which projects from the y Ergodic averages of arbitrary observablesP (y)A are given by

Z:(43 -2 d-dimensional phase space to a k-dimensional subspace. If ρ 10 1 −yHV2 I!"!!S!1F U!3"!!!FW 2 1 1 ␳y͑y͒ = e B 1!"+%%XW% y U!"!!−!RWy + O . ͑9͒ 2 @ !"RR#%R U!"!!X!S @H:7* ͩ 2 ͪ ͩ Nͪ d=k=1, the varianceA␴x of= this␳Gaussianx A x dx.is given by ͓8͔ 6 ͱ␲ ͫ ͱN ͬ

͗ ͑ ͒͘ ͑ ͒ ͑ ͒ Y* ͑ ͒ ͵ !"# −1 ϱ This result is in excellent agreement with numerical experi- 2 2 10-3 The average ␴͗x͘=vanishes.͗f͑x0͒ ͘ +For2͚the͗f͑correlationx0͒f͑xi͒͘. function one͑3͒ ments ͑Fig. 1͒. has i=1 We should mention at this point that the finite-N corrections are nonuniversal, i.e., different mappings have different Here ͗ ͘ denotes an expectation formed with the natural -4 ¯ 1 !"! finite-N corrections.10 For example, by applying the techniques invariant density of the ͗mapxi xi ͘T=. ␦i ,i . ͑7͒ 1 2 2 1 2 of ͓13,14͔ to the-4cubic map-2 0 2 4 As an example to illustrate these general results, let us ! y 3 considerDue tothethelogisticstrong mapmixing property, the conditions for the [*@H:7*Z:(43 xi+1 = 4xi − 3xiy, ͑10͒ z validity of a CLT are satisfied for a=22. This means the dis- Santa Fe Institute which has the same invariantSantadensity Fe Institute ͑5͒ as the logistic map tribution of the quantityxi+1 = T͑xi͒ = 1 − axi ͑4͒ FIG. 1. ͑Color online͒ Probability density of rescaled sums of withiteratesa=2ofbutthedifferlogisticent highermap-orderwithcorrelations,a=2 as givenwe obtainby Eq.in 8 , N ͑ ͒ on the interval X=͓−1,1͔. For1 a=2 the system is ͑semi͒con- leadingN=2ϫ10order6 and N=100. The number of initial values contributing to jugated to a Bernoulliyshiftª and͚strongly͑xi − ͗x͒͘mixing. The natural͑8͒ 6 7 ͱN the histogram is n1ini=22ϫ10 ,1respectively1 1 nini=10 . The solid lines invariant density is i=1 ␳ ͑y͒ = e−y 1 + y4 − y2 + . ͑11͒ correspondy to Eq.ͱ ͑9͒. ͫ Nͩ 12 4 16ͪͬ becomes Gaussian for N ϱ, regarding the initial value x ␲ → 1 1 as a random variable␳ withx =a smooth probability. distribution.5 Note that in this case the leading-order corrections are of x͑ ͒ 2 ͑ ͒ ͓13,14͔. Our final result is that for finite but large N the For the variance of this Gaussian␲ͱ1 − xwe obtain from Eqs. 3 ͑ ͒ orderprobability1/N, ratherdensitythanofofy isordergiven1/ͱbyN. Again our analytical and 7 the value ␴2 = 1 choosing f x =x . The above CLT Ergodic͑ averages͒ of arbitrary2 ͓ observables͑ ͒ A͔ are given by result is in good agreement with the numerics, see Fig. 2. result is highly nontrivial, since there are complicated Apparently the1 finite-−y2 N corrections1 3 to2 the asymptotic1 1 ␳y͑y͒ = e 1 + y − y + O . ͑9͒ higher-order correlations between the iterates of the logistic Gaussian behaviorͱ␲of dynamicalͫ ͱN systemsͩ 2 ͪsatisfyingͩ Nͪaͬ CLT map for a=2 ͑see͗A͓͑13x͔͒͘ =for͵details␳͑x͒͒A. ͑Thisx͒dxmeans. the ordinary͑6͒ are nonuniversal and can be used to obtain more information −1 CLT, which is only valid for independent xi cannot be di- onThistheresultunderlyingis in excellentdeterministicagreementdynamics.withThisnumericalis certainlyexperi- Therectlyaverageapplied,͗x͘anvanishes.extensionForof thetheCLcorrelationT for mixingfunctionsystemsoneis importantments ͑Fig.from1͒.a general physical point of view if the dy- hasnecessary ͓3͔. namicsWe underlyingshould mentiona CLTatis thisa prioripointunknown.that the finite-N correc- For physical and practical applications, the number N is tionsWe aremaynonuniversal,also look at i.e.,distributionsdifferentofmappingsthe variablehaveydifob-ferent always finite, hence it is important1 to know what the finite-N tainedfinite-forN corrections.“typical” parameterFor example,values byin theapplyingchaotictheregimetechniquesof ͗x x ͘ = ␦ . ͑7͒ corrections are for a giveni1 i2 dynamical2 i1,i2 system. This problem theof logistic͓13,14͔map,to thesuchcubicas amap=1.7,1.8,1.9. A CLT has not been can be solved for our example, the map T͑x͒=1−2x2, by rigorously proved in this case, but, nevertheless, we again 3 Dueapplyingto thethestronggeneralmixinggraph-theoreticalproperty, themethodsconditionsdevelopedfor thein observe Gaussian limit behaviorxi+1 = 4.xThisi − 3isxishown, in Fig. 3. It is͑10͒ validity of a CLT are satisfied for a=2. This means the dis- which has the same invariant density ͑5͒ as the logistic map tribution of the quantity 040106-2 with a=2 but different higher-order correlations, we obtain in N 1 leading order y ª ͚ ͑xi − ͗x͒͘ ͑8͒ ͱN 1 2 1 1 1 1 i=1 ␳ ͑y͒ = e−y 1 + y4 − y2 + . ͑11͒ y ͱ ͫ Nͩ 12 4 16ͪͬ becomes Gaussian for N ϱ, regarding the initial value x ␲ → 1 as a random variable with a smooth probability distribution. Note that in this case the leading-order corrections are of For the variance of this Gaussian we obtain from Eqs. ͑3͒ order 1/N, rather than of order 1/ͱN. Again our analytical 2 1 and ͑7͒ the value ␴ = 2 ͓choosing f͑x͒=x͔. The above CLT result is in good agreement with the numerics, see Fig. 2. result is highly nontrivial, since there are complicated Apparently the finite-N corrections to the asymptotic higher-order correlations between the iterates of the logistic Gaussian behavior of dynamical systems satisfying a CLT map for a=2 ͑see ͓13͔ for details͒. This means the ordinary are nonuniversal and can be used to obtain more information CLT, which is only valid for independent xi cannot be di- on the underlying deterministic dynamics. This is certainly rectly applied, an extension of the CLT for mixing systems is important from a general physical point of view if the dy- necessary ͓3͔. namics underlying a CLT is a priori unknown. For physical and practical applications, the number N is We may also look at distributions of the variable y ob- always finite, hence it is important to know what the finite-N tained for “typical” parameter values in the chaotic regime of corrections are for a given dynamical system. This problem the logistic map, such as a=1.7,1.8,1.9. A CLT has not been can be solved for our example, the map T͑x͒=1−2x2, by rigorously proved in this case, but, nevertheless, we again applying the general graph-theoretical methods developed in observe Gaussian limit behavior. This is shown in Fig. 3. It is

040106-2 Color Plates xix 6.4 Pictures of Strange Attractors and Fractal Boundaries 125

4.1 Parameter Dependence of the Iterates 35

r values are densely interwoven and one ﬁnds a sensitive dependence on parameter values. We also discuss the concept of structural universality and calculate the invariant density of the logistic map at r = 4. Finally, in Section 4.5 we present a summary that explains the parallels between the Feigenbaum route to chaos and ordinary equilibrium second-order phase transitions. This chapter ends with a discussion of the measurable properties of the Feigenbaum route and a review of some experiments in which this route has been observed. xx Color Plates xx Color Plates Plate VIII. 4.1 Parameter Dependence of the Iterates Interlude: fractals To provide an overview in this section, we present several results for the logistic map obtained by computerIn chaositeration theof eq. central(4.1) for different limitvalues distributionof the parameter r. theorFigure 23emshows holdsthe accumulation points of the iterates f n(x ) for n > 300 as a function of r together with the { r 0 } Liapunov exponent λ obtained via eq. (3.9).

Plate X.

Plate IX. Figure 94: Correspondence between the structure of “Mandelbrot’s set” M in the c-plane and the struc- What happens at the edge of 2 ture of bifurcations of the (transformed) logistic map xn+1 = xn + c along the real c-axis (after Peitgen and Richter, 1984). chaos? (zero Lyapunov exponent)

The set M is called “Mandelbrot’s set” after B. B. Mandelbrot who ﬁrst published Plate(1980)XI. a picture of M (see Fig. 94). It shows that M has also a fractal structure (but it is no Julia set). This study was extended by Peitgen and Richter (1984). If c does not belong to M, n then limn ∞ fc (0) ∞. Therefore, they deﬁne “level curves” in the following way: color → → a starting point according to the number of iterations it needs to leave a disk with a given radius R. As shown by Douady and Hubbard (1982), lines of equalPlate XI.color can be interpreted as equipotential lines if the set M is considered to be a charged conductor. Plates VIII–XV RAPID COMMUNICATIONS show the fascinating results of this procedure which brings us back to Ruelles’ remark at the beginning of this section. From “Deterministic Chaos”, Schuster & Just a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Santa Fe Institute Figure 23: Santa Fe Institute TIRNAKLI, BECK, AND TSALLIS PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒

prove the existence of a CLT. This means the probability 6 6 N=2x10 ; nini=2x10 6 distribution of N=100 ; nini=10x10 eq.(9) ; N=100 At zero Lyapunov exponent ? N 100 1 eq.(9) ; N=2x106 There are long (some times unconstructive and boring) debates on y ª f͑xi͒ ͑2͒ ͱN ͚ what mechanical statistical framework is appropriated to use at the so i=1 a=2 called: edge -or onset- of chaos. (For me,) more clear and exact is to 10-1 use, in this case, a less fancy expression: “at zero Lyapunov y becomes Gaussian for N ϱ, regarding the initial value x1 → d k )

exponent.” as a random variable. Here f : is a suitable smooth y (Sum) R →R ( function with vanishing average which projects from the y d-dimensional phase space to a k-dimensional subspace. If ρ 10-2 d=k=1, the variance ␴2 of this Gaussian is given by ͓8͔ ϱ 2 2 10-3 ␴ = ͗f͑x0͒ ͘ + 2͚ ͗f͑x0͒f͑xi͒͘. ͑3͒ i=1 Here denotes an expectation formed with the natural ͗¯͘ 10-4 invariant density of the map T. -4 -2 0 2 4 As an examplex0 (initialto illustrateconditionthese) general results, let us Santa Fe Institute consider the logistic map Santa Fe Institute y 2 xi+1 = T͑xi͒ = 1 − axi ͑4͒ FIG. 1. ͑Color online͒ Probability density of rescaled sums of iterates of the logistic map with a=2 as given by Eq. ͑8͒, on the interval X=͓−1,1͔. For a=2 the system is ͑semi͒con- N=2ϫ106 and N=100. The number of initial values contributing to jugated to a Bernoulli shift and strongly mixing. The natural 6 7 the histogram is nini=2ϫ10 , respectively nini=10 . The solid lines invariant density is correspond to Eq. ͑9͒. 1 ␳x͑x͒ = . ͑5͒ ␲ͱ1 − x2 ͓13,14͔. Our final result is that for finite but large N the probability density of y is given by Ergodic averages of arbitrary observables A are given by 1 −y2 1 3 2 1 1 ␳y͑y͒ = e 1 + y − y + O . ͑9͒ ͱ␲ ͫ ͱN ͩ 2 ͪ ͩ Nͪͬ ͗A͑x͒͘ = ͵ ␳͑x͒A͑x͒dx. ͑6͒ −1 This result is in excellent agreement with numerical experi- The average ͗x͘ vanishes. For the correlation function one ments ͑Fig. 1͒. has We should mention at this point that the finite-N corrections are nonuniversal, i.e., different mappings have different 1 finite-N corrections. For example, by applying the techniques ͗x x ͘ = ␦ . ͑7͒ i1 i2 2 i1,i2 of ͓13,14͔ to the cubic map 3 Due to the strong mixing property, the conditions for the xi+1 = 4xi − 3xi, ͑10͒ validity of a CLT are satisfied for a=2. This means the dis- which has the same invariant density ͑5͒ as the logistic map tribution of the quantity with a=2 but different higher-order correlations, we obtain in N 1 leading order y ª ͚ ͑xi − ͗x͒͘ ͑8͒ ͱN 1 2 1 1 1 1 i=1 ␳ ͑y͒ = e−y 1 + y4 − y2 + . ͑11͒ y ͱ ͫ Nͩ 12 4 16ͪͬ becomes Gaussian for N ϱ, regarding the initial value x ␲ → 1 as a random variable with a smooth probability distribution. Note that in this case the leading-order corrections are of For the variance of this Gaussian we obtain from Eqs. ͑3͒ order 1/N, rather than of order 1/ͱN. Again our analytical 2 1 and ͑7͒ the value ␴ = 2 ͓choosing f͑x͒=x͔. The above CLT result is in good agreement with the numerics, see Fig. 2. result is highly nontrivial, since there are complicated Apparently the finite-N corrections to the asymptotic higher-order correlations between the iterates of the logistic Gaussian behavior of dynamical systems satisfying a CLT map for a=2 ͑see ͓13͔ for details͒. This means the ordinary are nonuniversal and can be used to obtain more information CLT, which is only valid for independent xi cannot be di- on the underlying deterministic dynamics. This is certainly rectly applied, an extension of the CLT for mixing systems is important from a general physical point of view if the dy- necessary ͓3͔. namics underlying a CLT is a priori unknown. For physical and practical applications, the number N is We may also look at distributions of the variable y ob- always finite, hence it is important to know what the finite-N tained for “typical” parameter values in the chaotic regime of corrections are for a given dynamical system. This problem the logistic map, such as a=1.7,1.8,1.9. A CLT has not been can be solved for our example, the map T͑x͒=1−2x2, by rigorously proved in this case, but, nevertheless, we again applying the general graph-theoretical methods developed in observe Gaussian limit behavior. This is shown in Fig. 3. It is

040106-2 RAPID COMMUNICATIONS

TIRNAKLI, BECK, AND TSALLIS PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒ Limit distribution at zero Lyapunov exponent Self-similarity behavior prove the existence of a CLT. This means the probability N=2x106 ; n =2x106 ini (and near zero Lyap.6 exponent) distribution of N=100 ; nini=10x10 eq.(9) ; N=100 N 100 1 eq.(9) ; N=2x106 y ª f͑xi͒ ͑2͒ ͱN ͚ i=1 a=2 10-1 becomes Gaussian for N ϱ, regarding the initial value x1 → d k )

as a random variable. Here f : is a suitable smooth y

R →R (

function with vanishing average which projects from the y d-dimensional phase space toPa (k-dimensionaly) subspace. If ρ 10-2 d=k=1, the variance ␴2 of this Gaussian is given by ͓8͔ ϱ 2 2 10-3 ␴ = ͗f͑x0͒ ͘ + 2͚ ͗f͑x0͒f͑xi͒͘. ͑3͒ i=1 Here denotes an expectation formed with the natural ͗¯͘ 10-4 invariant density of the map T. -4 -2 0 2 4 As an example to illustrate these general results, let us consider the logistic map y 2 xi+1 = T͑xi͒ = 1 − axi ͑4͒ FIG. 1. ͑Color online͒ Probability density of rescaled sums of iterates of the logistic map with a=2 as given by Eq. ͑8͒, on the interval X= −1,1 . For a=2 the system is semi con- ͓ ͔ ͑ ͒ N=2ϫ106 and N=100. Theynumber of initial values contributing to jugated to a Bernoulli shift and strongly mixing. The natural 6 7 the histogram is nini=2ϫ10 , respectively nini=10 . The solid lines invariant density is correspond to Eq. ͑9͒.

Santa Fe Institute 1 Santa Fe Institute ␳x͑x͒ = . ͑5͒ ␲ͱ1 − x2 ͓13,14͔. Our final result is that for finite but large N the probability density of y is given by Ergodic averages of arbitrary observables A are given by 1 −y2 1 3 2 1 1 ␳y͑y͒ = e 1 + y − y + O . ͑9͒ ͱ␲ ͫ ͱN ͩ 2 ͪ ͩ Nͪͬ ͗A͑x͒͘ = ͵ ␳͑x͒A͑x͒dx. ͑6͒ −1 This result is in excellent agreement with numerical experi- The average ͗x͘ vanishes. For the correlation function one ments ͑Fig. 1͒. has We should mention at this point that the finite-N corrections are nonuniversal, i.e., different mappings have different 1 finite-N corrections. For example, by applying the techniques ͗x x ͘ = ␦ . ͑7͒ Things we have discussed: i1 i2 2 i1,i2 of ͓13,14͔ to the cubic map 3 Due to the strong mixing property, the conditions for the xi+1 = 4xi − 3xi, ͑10͒ validity of a CLT are satisfied for a=2. This means the dis- Chaos and uncertainty which has the same invariant density ͑5͒ as the logistic map tribution of the quantity with a=2 but different higher-order correlations, we obtain in N Deterministic variables behave as random variables 1 leading order y ªPatter͚ ͑xnsi − ͗x͒͘ ͑8͒ ͱN 1 2 1 1 1 1 i=1 ␳ ͑y͒ = e−y 1 + y4 − y2 + . ͑11͒ y ͱ ͫ Nͩ 12 4 16ͪͬ Simple systems are amazingly rich and there are still important open becomes Gaussian for N ϱ, regarding the initial value x ␲ → 1 questions in its behavior as a random variable withMiguela smooth Angelprobability Fuentes distribution. Note that in this case the leading-order corrections are of For the variance of thisSantaGaussian Fe Institute,we obtain USAfrom Eqs. ͑3͒ order 1/N, rather than of order 1/ͱN. Again our analytical 2 1 and ͑7͒ the value ␴ =Instituto2 ͓choosing Balseirf͑x͒=ox and͔. The CONICETabove CL,T Argentinaresult is in good agreement with the numerics, see Fig. 2. result is highly nontrivial, since there are complicated Apparently the finite-N corrections to the asymptotic higher-order correlations between the iterates of the logistic Gaussian behavior of dynamical systems satisfying a CLT map for a=2 ͑see ͓13͔ for details͒. This means the ordinary are nonuniversal and can be used to obtain more information CLT, which is only valid for independent xi cannot be di- on the underlying deterministic dynamics. This is certainly rectly applied, an extension of the CLT for mixing systems is important from a general physical point of view if the dy- necessary ͓3͔. namics underlying a CLT is a priori unknown. For physical and practical applications, the number N is We may also look at distributions of the variable y ob- always finite, hence it is important to know what the finite-N tained for “typical” parameter values in the chaotic regime of corrections are for a given dynamical system. This problem the logistic map, such as a=1.7,1.8,1.9. A CLT has not been Santa Fe Institute can be solved for our example, the map T͑x͒=1−2x2, by rigorously proved in this case, but,Sanevertheless,nta Fe Institute we again applying the general graph-theoretical methods developed in observe Gaussian limit behavior. This is shown in Fig. 3. It is

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• Waves and excitable media

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• Beyond standard analysis

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Linear stability analysis PDE, Reaction Diffusion Models:

(u0, v0): homogenous steady state, i. e.: “Under certain conditions spatially inhomogeneous patterns can evolve by diffusion driven instability” A. M. Turing, 1952 f(u0, v0) = g(u0, v0) = 0

The chemical basis of morphogenesis, A. M. Turing, Phil. Trans. Roy. Soc. London, B237, 37-72, 1952. u u w = 0 Minimal model v − v ! − 0 " 2 ∂tu = f(u, v) + du u fu fv ∇ w˙ = Aw A = 2 gu gv ∂ v = g(u, v) + d v ! " t v∇ λ > 0 w w eλt → ∞ ∼ λ < 0 w 0 → Santa Fe Institute { Santa Fe Institute Conditions for stability of the homogeneous state Conditions for instability of the homogeneous state

The eigenvalues are given by the solution of 2 ∂tu = f(u, v) + du u ∇2 f λ f ∂tv = g(u, v) + dv v u v = 0 ∇ g− g λ ! u v − ! ! ! ! ! Linear version Then there are two solution: λ1, λ2 from the equation:! ! 2 λt ∂tw = Aw + D w w = cke Wk(r) ∇ k λ2 (f + g )λ + (f g f g ) = 0 ! − u v u v − v u Using cos functions Linear stability, [ λ ] < 0 ,is guaranteed if ! λW = AW + D 2W k k ∇ k = AW Dk2W f f k k A = u v − trA = fu + gv < 0, A = fugv fvgu > 0 g g | | − ! u v "

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Eigenvalues equation Conditions

λt λI A + Dk2 = 0 w = cke Wk(r) | − | Stability of the homogeneous state !k trA = f + g < 0, A = f g f g > 0 u v | | u v − v u 2 2 2 λ + λ[k (1 + d) (fu + gv)] + h(k ) = 0 − Instability of some modes 2 2 4 2 dfu + gv > 0, (dfu + gv) 4d A > 0 h(k ) = dk (dfu + gv)k + A − | | | |

Linear stability, [ λ ] > 0 ,is guaranteed if λ λ2 + λ[k2(1 + d) (f + g )] + h(k2) = 0 ! − u v

k df + g > 0, (df + g )2 4d A > 0 u v u v − | |

Santa Fe Institute Santa Fe Institute Discussion Waves

Easy frame work Minimal non linear model with diffusion

Two diffusion coefﬁcient ∂tu = ku(1 u) + D∂xxu Fisher, 1937 Extensive literature −

∂ u = u(1 u) + ∂ u t − xx

Static frame

u(x, t) = U(z), z = x ct −

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Linear stability analysis

stable node if c2 4 2 1/2 ≥ U !! + cU ! + U(1 U) = 0 (0, 0) : λ = [ c (c 4) ]/2 − ± − ± − stable spiral if c2 < 4 (1, 0) : λ = [ c (c2 + 4)1/2]/2 { lim U(z) = 0, lim = 1 ± − ± z z →∞ →−∞

In the plane (U, V )

U ! = V, V ! = cV U(1 U) − − −

c c = 2√kD ≥ min

Santa Fe Institute Santa Fe Institute Discussion Excitable media

Finite domain: localized structure

ut = f(u, v), vt = g(u, v) Critical Size

Nullclines

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Excitable media with space Spiral waves: self sustained spatiotemporal structures

Refractory zone ∂tu = f(u, v) + ∂xxu ∂tv = g(u, v) Directions of the front

Excitable state

Activated zone c

Refractory state

Santa Fe Institute A. S. Mikailov, Foundation of Synergetics Santa Fe Institute More dimensions Discussion

Analytical method for velocity, geometrical properties of the spiral, etc.

Multi-arms spirals

Interaction of spirals

Cellular automata methods

A. S. Mikailov, Foundation of Synergetics

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Non local mechanisms

∂tu = f(u) + g(u) G(x, x!) u(x!) dx!, G = 1 !Ω !Ω

∂tu = f(u) + g(u) G(x, x!) u(x!) dx!, G = 1 Ω Ω ! ! Near the stable point

∂tu = f !(u0)u + g!(u0)u0u + g(u0) G u dx!, u0 is a stable point !

Santa Fe Institute Santa Fe Institute Example Doing the Fourier transform, i. e. u eikx+λt ∼ 2 ∂ u = f(u) + g(u) G(x, x!) u(x!) dx! + d u t ∇ !Ω λ = f (u ) + g (u )u + g(u )F (k), with F (k) = Geikydy f(u) = au, g(u) = bu ! 0 ! 0 0 0 − ! θ[w (x x!)]θ[w + (x x!)] G(x, x!) = − − − Equation with space variables 2w w +w − 2 ∂tu = f(u) + g(u) G(x, x!) u(x!) dx! + d u Note: G δ , is the Fisher equation Ω ∇ ! 10506 J. Phys. Chem.→B, Vol. 108, No. 29, 2004 Unstable Fuentesmodeset al. Stable steady-state patterns require that

D λ > 2π (6) Condition for unstable modes -aF(λ) sin(kw) 2 λ whereλ =λ ) 2π/k ais the wavelength" associateddkwith the k mode of the Fourier expansion of the pattern. Condition−(6) provideska wnecessary condition− to check for 2 the existence or absence of inhomogeneity, i.e., patterns, in the Dk f !(u0) g!(u0)u0 steady state. We see from (6) that the Fourier transform of the F (k) > − − influence function at the wavelength under consideration should g(u0) be negatiVe for the patterns to appear and that its magnitude should be large enough. One way of understanding this condition is to recast it as requiring that 2π times the “effective diffusion length” should be smaller than the wavelength for the patterns to occur. By the effective diffusion constant, we mean D divided Figure 1. Dispersion relation (9) betweenkthe dimensionless growth by -F(λ), which is a factor decided by the influence function, exponent %! and wavenumber k! plotted for different values of the ratio and by the diffusion length, we mean the distance traversed η of the influence function range to the diffusion length (see the text). Santa Fe Institute diffusively in a time interval on the order of the inverse of the Values of η are 50 (solid line), 10 (dashed line), and 2 (dotted line).Santa Fe Institute growth rate. If the influence function is smooth such as in the Patterns appear for those values of k! for which % is positive. case of a Gaussian in an infinite domain, the Fourier transform The earlier finding1,2 that no patterns appear for extremes of is positive and no patterns appear. A cutoff in the influence the range of the influence function is clear from eq 9. As the function produces oscillations in the Fourier transform, which influence width vanishes, i.e., as η goes to zero, both terms in can go negative for certain wavelengths. The reported finding1 % are negative and there can be no steady-state patterns: we that the cutoff nature appears necessary to pattern formation recover the solution for the local limit, corresponding to eq 1, can be understood naturally in this way. when w f 0. Because the boundary conditions are periodic in Specific Examples a domain of length L, there are only the allowed values k ) nπ/L of the wavenumber. Therefore, in the opposite limit of Let us consider, in turn, three cases of the influence function full range, i.e., w f L, the sine term vanishes, %! ) -k!, and that we have used in our earlier investigations:1 square, cutoff again there are no patterns. Gaussian, and intermediate. Precisely, the same qualitative behavior occurs for other First we take nonsquare influence functions such as the Gaussian with a Traveling waves - Non local Interactions cutoff, i.e., for 1 f(x - y) ) {θ[w - (x - y)] θ[w + (x - y)]} (7) 2w 1 x - y 2 Frontiers f(x - y) ) exp - {θ[w - (x - y)] σ π erf(w/σ) [ ( σ ) ] where θ is the Heaviside function. The influence function is " thus a square of the cutoff range measured by w from its center. θ[w + (x - y)]} (10) We will consider the case here when the range w is smaller We again consider the case when the cutoff length does not than, or equal to, the domain length L. Equation 4 then involves exceed the domain length. This leads to the Fourier transform an integral from 0 to w and gives 1- Lacking of ﬁrst principles descriptionsof the influence function involving an integral from 0 to w. In sin(kw) these as well as other cases considered, it should be appreciated % ) -a - Dk2 (8) that the domain length L, if taken to be smaller than the cutoff θ[x]θ[w + (x x!)] kw length, becomes itself the cutoff length: factors such as kw appearing in the Fourier transform become then kL instead. G(x, x!) = − In terms of dimensionless parameters w 0 +w For this cutoff Gaussian case, the square case dispersion %! ) %/a relation (8) is replaced by 2 2- A different geometry a exp[-(kσ/2) ] w ikσ w ikσ k! ) k D/a % ) - erf - + erf + - Dk2 " 2 erf(w/σ) [ (σ 2 ) (σ 2 )] η ) w a/D " The dimensionless version (9) is replaced by we have exp(-k!&2) %! ) - {erf(R - ik!&) + erf(R + ik!&)} - k! sin(k!η) 2 erf(R) sin(kw) %! ) - - k! (9) (11) Patters k!η [F (k)] = Here R and 2& are the ratios of the cutoff length to the range which we plot in Figure 1 for three different values (50, 10, ! kw and of the range to the diffusion length respectively: and 2) of the ratio η of the width to the diffusion length (not effectiVe diffusion length). For the third case, there are no R ) w/σ patterns: diffusion is strong enough to wash them out. For the 1 sin(kw) intermediate case, patterns can occur with wavelengths corre- 2& ) σ a/D Oscillations sponding to values of around k! 0.4, while for the η ) 50 " [F (k)] = = 0 ≈ case, they occur around k! 0.1. What is analogous to η in the square case is their product 2R& " kw − kw $ ≈

Santa Fe Institute Santa Fe Institute Lacking of ﬁrst principles descriptions A different geometry

Example Example Non continuos Systems (hoping for an approach) Granular system The Sierpinski gasket (and its wave equation)

∂2u = 2u t ∇ Non continuos operator 2 3 m ∂t u = lim 5 Hmu, Hmu = O(x, y) Roughly speaking m 2 cos(ωt) →∞ Yamaguti et al., Translations of mathematical monographs, American Mathematical Society, 1997 Markovian { Different solutions (eigenfunctions) Non Markovian, memory

{ Not only different patterns, but physics etc

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Conclusions

• Linear stability analysis is a fundamental tool Using a similar approach is my hope that in networks studies where we can deﬁne a • Turing patterns has many application, we must have a critic view

geometry -distances, etc- it will be possible • In the literature: studies for ‘amplitude equations’ to deﬁne a : “Calculus on network” • Non continuos systems need special attention

• Dynamical systems knowledge is also necessary

• The geometry where the system lives is very important

• Calculus on network

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