Conference on Complex Systems -Foundations and Applications- 30 November 2013, Rio de Janeiro, Brasil in collaboration with : Constantino Tsallis (CBPF, Brasil)Christian Beck (London U., UK)zgr Afar (Ege University)Central Limit Behaviour of Dynamical Systems: Emergence of q-Gaussians and Scaling Laws

OUTLINE of the TALKCentral Limit Theorem (CLT)2) CLT for 1d Discrete Dynamical Systems Results of the Logistic map at fully developed chaotic pointResults of the Logistic Map at other chaotic points Results of the Logistic Map at the edge of chaos Closer Look for the Logistic Map at the edge of chaosScaling Laws for the Logistic Map at the edge of chaos3) Conclusions

Central Limit Theorem (CLT)CLT is a very important concept in probability theory and it plays a fundamental role in statistical physics.Basically CLT states that the sum of N independent identicallydistributed random variables, appropriately rescaled and centered, has a Gaussian distribution in the limitNamely,xi : random variables 2 : variancef(xi) : a suitable smooth function.

Central Limit Theorem (CLT)(deterministic dynamical systems)There are also CLTs for the iterates of deterministic dynamical systems. M. Kac, Ann. Math. 47 (1946) 33. M.C. Mackey and M. Tyran-Kaminska, Phys. Rep. 422 (2006) 167. P. Billingsley, Convergence of Probability Measures (Wiley, 1968).But, if the assumption of independent identically distributed is replaced by the property that the dynamical system is sufficiently strongly mixing, then various CLTs can be proved for deterministic dynamical systems. The iterates of such systems can never be completely independent.

Central Limit Theorem (CLT)(deterministic dynamical systems)We are interested in two fundamentally important questions :Suppose a CLT is valid for a deterministic system for ; what are the leading-order corrections to the CLT for finite N ? Question 1 : Suppose the dynamical system is not sufficiently mixing and it does not satisfy a standard CLT ; what are typical distributions for these systems in the limit ? Question 2 : ? See for details: U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).

whereac=1.40115518909205...lCLT(logistic map)chaoticperiodic

Central Limit Theorem (CLT)(logistic map)If f is sufficiently strongly mixing, one can prove the existence of a CLT, namely the probability distributionbecomes Gaussian for and the variance is given by For the fully developed chaotic state of logistic map (a=2)with This CLT result is highly nontrivial since there are complicated higher-order correlations between the iterates.

Central Limit Theorem (CLT)(logistic map at a=2 point)U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).a=2N=2x106nini=2x106Logistic map (a=2 case)

Central Limit Theorem (CLT)(logistic map at other chaotic points)U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).Logistic map (other chaotic cases)l a=1.7l a=1.8l a=1.9N=2x106nini=2x106The variancel s2=0.0186l s2=0.1248l s2=0.0613

If random variables are independent and variance is finite , for , a Gaussian attractor emerges for .

If random variables are independent and variance is diverging , for , an a-stable attractor (Levy distribution) emerges for .

Central Limit Theorem (CLT)xi : random variablesprobability distribution function (conveniently centered and scaled)

whereLogistic map at the edge of chaosAt the edge of chaos :U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).Logistic map (a=ac=1.40115518909...)N=215nini=16x106q=1.75 ; b=13q-Gaussiannot a Gaussian.!!!One of the candidate functions is,

If random variables are independent and variance is finite , for , a Gaussian attractor emerges for .

If random variables are independent and variance is diverging , for , an a-stable attractor (Levy distribution) emerges for .

If random variables are globally correlated and suitable variance is finite , for , a q-Gaussian attractor emerges for .

If random variables are globally correlated and suitable variance is diverging , for , a (q,a)-stable attractor emerges for . Central Limit Theorem (CLT)xi : random variablesprobability distribution function(conveniently centered and scaled)S. Umarov, C. Tsallis and S. Steinberg, Milan J. Math. 76 (2008) 307S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, J. Math. Phys. 51 (2010) 033502 C. Vignat and A. Plastino, J. Phys. A 40 (2007) F969M. Hahn, X. Jiang and S. Umarov, J. Phys. A 43 (2010) 165208

Logistic map at the edge of chaos(closer look)At the edge of chaos, for a full description of the shape of the distribution function on the attractor, two ingredients are necessary :U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 0562091) Precision of ac2) Number of iterations NFor a given finite precision of ac , if N is very large, then the system feels that it is not exactly at the edge of chaos and the central part of its probability distribution function becomes a Gaussian (with small deviations in the tails).For a given finite precision of ac , if N is too small, then the summation starts to be inadequate to approach the edge of chaos limiting distribution and the central part of the distribution exhibits a sort of divergence.

Logistic map at the edge of chaos(closer look)N=215nini=4x106N=219N=212GaussianSo, if and

if you take N=219 if you take N=212 Then, what is the optimum value of N ? (in order to achieve best convergence to the limit distribution)very large too small U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

Logistic map at the edge of chaos(closer look)An attempt for a theoretical argument on the optimum value of N :

Suppose we are slightly above the critical point by an amountThen there exists 2n chaotic bands of the attractor, which approach the Feigenbaum attractor for by the band splitting procedure. So, after 2n iterations, the sum of the iterates will approach a fixed value plus a small correction , which describes the small fluctuations of the position of the 2nth iterate within the chaotic band. Therefore, where d=4.669 is the Feigenbaum constant)If we continue with another 2n iterations, then another 2n iterations , after 22n iterations we get the total sum of iterates as This can be regarded as a sum of 2n strongly correlated random variable , each being influenced by the structure of the 2n chaotic bands at distance from the Feigenbaum attractor.

Logistic map at the edge of chaos(closer look)In the frame of this scaling argument, the optimum value of N (say N*) to observe convergence to the limit distribution is given bywhere, at a given distance , the number comes fromU. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209Afterwards, we noticed that this scaling argument is a direct consequence of famous Huberman-Rudnick scaling law

Huberman-Rudnick scaling lawO. Afsar and U. Tirnakli, EPL 101 (2013) 20003.

Huberman-Rudnick scaling law. Afar and U. Tirnakli, EPL 101 (2013) 20003.

Logistic map at the edge of chaos(closer look)In order to check this argument, we numerically study various values given in the Table.U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

Logistic map at the edge of chaos(closer look)U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

ac=1.4011551890920505N*=254 ~ 1.8x1016 !!!Logistic map at the edge of chaos(closer look)N=215

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Logistic map at the edge of chaos(scaling laws)O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

O. Afsar and U. Tirnakli, Relationships and Scaling Laws Among Correlation, Fractality, Lyapunov Divergence and q-Gaussian Distributions, (2013), preprint.

Conclusions

At the edge of chaos, we show that the relevant limit distributions appear to be q-Gaussians for these systems.

These results represent a kind of power-law generalization of the CLT.

We obtain several novel scaling laws relating the range of q-Gaussians with the correlation length, fractal dimension and Lyapunov exponent.

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