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Cellular automata are fully discrete complex systems with parallel and homogeneous behavior studied both from the theoretical and modeling viewpoints. The limit behaviors of such systems are of particular interest, as they give insight into their emerging properties. One possible approach to investigate such limit behaviors is the analysis of the growth of graphs describing the finite time behavior of a rule in order to infer its limit behavior. Another possibility is to study the Fourier spectrum describing the average limit configurations obtained by a rule. While the former approach gives the characterization of the limit configurations of a rule, the latter yields a qualitative and quantitative characterisation of how often particular blocks of states are present in these limit configurations. Since both approaches are closely related, it is tempting to use one to obtain information about the other. Here, limit graphs are automatically adjusted by configurations directly generated by their respective rules, and use the graphs to compute the spectra of their rules. We rely on a set of elementary cellular automata rules, on lattices with fixed boundary condition, and show that our approach is a more reliable alternative to a previously described method from the literature.

One way to characterize an orbit of a dynamical system is through its frequency content. Using a “spectral bifurcation diagram”, it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation.

This study presents the application of a new method for generating synthetic accelerograms based on statistical distributions for Fourier phase differences and Fourier amplitudes as functions of earthquake magnitude, hypocentral distance and site geology. Two important characteristics of the methodology are that it requires a small number of input parameters and that ground motion time histories can be simulated without any specific modulation function. Two areas with different tectonic patterns (North-Eastern and Central Italy) were selected for the application. The results of our analysis are reliable in the case of Central Italy because the data set is large and quite uniformly distributed, while for North-Eastern Italy our results should not be used for distances greater than 30 km.

A theoretical attenuation model of earthquake-induced ground motion is presented and discussed. This model is related directly to physical quantities such as source and wave motion parameters. An attenuation formula for rms acceleration of ground motion is derived and verified using acceleration data from moderate-sized earthquakes recorded in Iceland from 1986 to 1997. The source parameters and the crustal attenuation are computed uniformly for the applied earthquake data. Furthermore, attenuation formulas for peak ground acceleration are put forward.

Some seismic design codes require to determine earthquake motions at ground surface, although we need sometimes a motion at engineering basement of ground. For example, it is necessary for optimal design of structures to introduce effects of soil-structure interaction and to input ground motions to engineering basement, which should satisfy the motions at surface required by a design code. This is simplified as a problem to find a possible input signal of a known non-linear-single-input and single-output (SISO) system with a given output signal. This study proposes a very simple algorithm to find a possible input signal for a known SISO system satisfying a given output signal from the system. The proposed algorithm searches for a possible input signal, whose system response is similar in shape to the given output signal, without any system identifications. Thus, it is not an inversion technique; it is an algorithm used to search for one of the possible input signals. Herein, the proposed algorithm is described and its performance is demonstrated using numerical examples.

As the original definition on Hilbert spectrum was given in terms of total energy and amplitude, there is a mismatch between the Hilbert spectrum and the traditional Fourier spectrum, which is defined in terms of energy density. Rigorous definitions of Hilbert energy and amplitude spectra are given in terms of energy and amplitude density in the time-frequency space. Unlike Fourier spectral analysis, where the resolution is fixed once the data length and sampling rate is given, the time-frequency resolution could be arbitrarily assigned in Hilbert spectral analysis (HSA). Furthermore, HSA could also provide zooming ability for detailed examination of the data in a specific frequency range with all the resolution power. These complications have made the conversion between Hilbert and Fourier spectral results difficult and the conversion formula is elusive until now. We have derived a simple relationship between them in this paper. The conversion factor turns out to be simply the sampling rate for the full resolution cases. In case of zooming, there is another additional multiplicative factor. The conversion factors have been tested in various cases including white noise, delta function, and signals from natural phenomena. With the introduction of this conversion, we can compare HSA and Fourier spectral analysis results quantitatively.