Behaviors of low-dimensional dynamical systems at vincinity of the chaos threshold and their robustness analysis

In this thesis we numerically investigate the distribution of the sums of the iterates of the logistic map, which is a good example for a dimensional dissipative dynamical system, and the relationships among the important properties of the nonlinear dynamics in the vicinity of the chaos threshold by adding two kinds of contributions with different densities. The first one is the well-known white noise, whereas the second is a newly defined one, named as correlated noise, which makes contributions from the own structure of the map. As the chaos threshold is approached, the iterates of the standard logistic map (i.e noise-free) have strong correlations and the standard Central Limit Theorem is not valid anymore. In a recent work (Tirnakli et al. (2009), it has been shown that the limit distribution seems to converge to q-Gaussian distribution, which maximizes the nonadditive entropy "S" _"q" "≡" (("1 - " ∑_"i" ▒"p" _"i" ^"q" ))⁄(("q - 1" ) ) under approapriate conditions. In this thesis, we investigate the effect of these contributions (i.e white noise and correlated noise) on the limit distribution and on the range of the obtained q-Gaussian distribution. As a result of these findings, under the existence of white noise and also the correlated, we analyze the scaling relations among correlation, fractality, the Lyapunov divergence and q-Gaussian distributions.