U g ur TIRNAKLI Ege U niversit y , F aculty of Science , Dept . of Physics , İzmir - Turkey

1. Ugur TIRNAKLI EgeUniversity, Faculty ofScience, Dept. of Physics, İzmir - Turkey Central Limit Behaviour of Dynamical Systems:Emergence of q-Gaussians and Scaling Laws in collaboration with : Constantino Tsallis(CBPF, Brasil) Christian Beck (London U., UK) Özgür Afşar (Ege University) Conference on Complex Systems -Foundations and Applications- 30 November 2013, Rio de Janeiro, Brasil

2. OUTLINE of the TALK • Central Limit Theorem (CLT) 2) CLT for 1d Discrete Dynamical Systems ğResults of the Logistic map at fully developed chaotic point ğResults of the Logistic Map at other chaotic points ğResults of the Logistic Map at the edge of chaos ğCloser Look for the Logistic Map at the edge of chaos ğScaling Laws for the Logistic Map at the edge of chaos 3) Conclusions

3. 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 Nindependent identically distributed random variables, appropriately rescaled and centered, has a Gaussian distribution in the limit . Namely, xi: random variable s2: variance f(xi): a suitable smooth function

4. Central Limit Theorem (CLT) (deterministic dynamical systems) There are also CLTs for the iterates of deterministic dynamical systems. The iterates of such systems can never be completely independent. 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. • 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).

5. Central Limit Theorem (CLT) (deterministic dynamical systems) We are interested in two fundamentally important questions : Question 1 : Suppose a CLT is valid for a deterministic system for ; See for details: U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R). what are the leading-order corrections to the CLT for finite N ? Question 2 : 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 ? ?

6. CLT (logistic map) where ac=1.40115518909205... chaotic l periodic

7. Central Limit Theorem (CLT) (logistic map) If f is sufficiently strongly mixing, one can prove the existence of a CLT, namely the probability distribution becomes 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.

8. Central Limit Theorem (CLT) (logistic map at a=2 point) a=2 N=2x106 nini=2x106 Logistic map (a=2 case) U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).

9. Central Limit Theorem (CLT) (logistic map at other chaotic points) la=1.7 la=1.8 la=1.9 N=2x106 nini=2x106 Logistic map (other chaotic cases) The variance ls2=0.0186 ls2=0.1248 ls2=0.0613 U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).

10. Central Limit Theorem (CLT) xi: random variables probability distribution function (conveniently centered and scaled) Ifrandomvariablesareindependentandvariance isfinite, for , a Gaussianattractoremergesfor . Ifrandomvariablesareindependentandvariance is diverging, for , an a-stableattractor (Levydistribution) emergesfor .

11. Logistic map at the edge of chaos At theedge of chaos : not a Gaussian.!!! One of the candidate functions is, q=1.75 ; b=13 where N=215 nini=16x106 q-Gaussian Logistic map (a=ac=1.40115518909...) U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106(R).

12. Central Limit Theorem (CLT) xi: random variables probability distribution function (conveniently centered and scaled) 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 isfinite , for , a q-Gaussian attractor emerges for . If random variables are globally correlated and suitable variance isdiverging , for , a (q,a)-stable attractor emerges for . S. Umarov, C. Tsallis and S. Steinberg, Milan J. Math. 76 (2008) 307 S. 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) F969 M. Hahn, X. Jiang and S. Umarov, J. Phys. A 43 (2010) 165208

13. 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 : 1) Precision of ac 2) Number of iterations N For 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. U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

14. Logistic map at the edge of chaos (closer look) So, if and if you take N=219 N=219 N=212 Gaussian very large N=215 nini=4x106 if you take N=212 too small Then, what is the optimum value of N ? (in order to achieve best convergence to the limit distribution) U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

15. 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 amount where d=4.669… is the Feigenbaum constant) Then 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, 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.

16. 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 by where, at a given distance , the number comes from Afterwards, we noticed that this scaling argument is a direct consequence of famous Huberman-Rudnick scaling law U. Tirnakli, C. Tsallis, C. Beck, PRE 75 (2009) 056209

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

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

19. 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

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

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

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

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

30. 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.