Dynamics Of Complex System And Scaling Behaviors Of The Interface In Random Media

In this thesis, we mainly investigate dynamics of the complex system and the scaling behavior of the evolving interface in random media.For the first time, we study the local chaotic properties of the Bak-Sneppen (BS) model. The local Lyapunov exponent describes the divergent degree of the evolving trajectories for the dynamical systems under external perturbations. The BS model is a model for complex system. There are two difficulties in investigating the chaotic properties of the BS model: One is how to define the separate degree of the two evolution trajectories. The other is how to deal with the intrinsic randomness in the evolution progress of the system. Our definition of the local Lyapunov exponent (λ) for the BS model solves the above two difficulties. It can also be used to analyze the chaotic properties of the BS model effectively. We findλis large in the early stage when the one- or two-dimensional BS model evolves to the self-organized critical (SOC) state, then decreases gradually to a minimum value, and increases to a saturation finally. This means that the BS model has a high regularity during the evolution process in the early stage. The appearance of the minimum ofλinfers the structure stability in the BS model. The following increase to approach a saturation value ofλreflects the obvious chaotic characters of the system under the self-organized critical state. The values ofλfor the one- and the two-dimensional BS system are with the same order of magnitude. Moreover, the saturation values ofλfor the two models are similar. This means that the dimension of the system has little effect on the local chaotic properties of the model. It is consistent with the fact that appearance of the chaos for a complex system in the self-organized critical state is not a true but on the edge of chaos. In such a state the system will become either pure chaos or pure order under perturbation.We systematically analyzes the fluctuations and correlations in the BS models. The Detrended Fluctuation Analysis (DFA) method is first applied to study the fluctuations and correlations in the complex system. With three typical characters in the BS model, the minimum fitness, the position of the minimum fitness and the number of active sites with fitness below the gap, as temporal series, we study the scaling property of the root mean square (rms) of the detrended fluctuations with the DFA method. We find that the rms of the detrended fluctuations for the series of minimum fitness increases with the size of boxes in a power-law, indicating existence of stable weak long-range scale-free correlation among the values of the minimum fitness. The rms fluctuations for the location of minimum site and the number of active sites are similar in behavior. The non-power-law behavior shows that the correlation changes with the windows size. Under small scales, the slope is about 0.5, indicating a random change with extremely weak short range correlation. Under large scales, the rms fluctuations saturate. Therefore, there are strong negative long-range correlations to ensure stationarity of the system at large time scales. The power spectrum of the series of the number of active sites shows a 1/f noise behavior.Kardar-Parisi-Zhang (KPZ) equation is an essential equation describing the growth of interfaces in the random media. An analytical expression for the scaled width has been deduced in a rigorous way from the KPZ equation. In our expression, the scaled roughness (?) turns out to depend on two scaling variables instead of separately depending on five parameters in the equation, as naively expected. Numerical calculations show that any variable of the five parameters in the KPZ equation has no effect on the scaling function (?) if the two scaling variables are fixed,. In addition, we also deduce the scaling function for the roughness from the Edward-Wilkinson (EW) equation. We find that this function can exactly describe the scaling behavior of the (?) for the one dimensional KPZ equation. Whereafter, the frontier progress in the study of the dynamics of complexity and interface growth in the random media are briefly introduced as well. At the end of this thesis, we especially introduce the new theoretical method in the study of the interface growth in the random media.