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Recurrence plots, a rather promising tool of data analysis, have been introduced by Eckman et al. in 1987. They visualise recurrences in phase space and give an overview about the system's dynamics. Two features have made the method rather popular. Firstly they are rather simple to compute and secondly they are putatively easy to interpret. However, the straightforward interpretation of recurrence plots for some systems yields rather surprising results. For example indications of low dimensional chaos have been reported for stock marked data, based on recurrence plots. In this work we exploit recurrences or ``naturally occurring analogues'' as they were termed by E. Lorenz, to obtain three key results. One of which is that the most striking structures which are found in recurrence plots are hinged to the correlation entropy and the correlation dimension of the underlying system. Even though an eventual embedding changes the structures in recurrence plots considerably these dynamical invariants can be estimated independently of the special parameters used for the computation. The second key result is that the attractor can be reconstructed from the recurrence plot. This means that it contains all topological information of the system under question in the limit of long time series. The graphical representation of the recurrences can also help to develop new algorithms and exploit specific structures. This feature has helped to obtain the third key result of this study. Based on recurrences to points which have the same ``recurrence structure'', it is possible to generate surrogates of the system which capture all relevant dynamical characteristics, such as entropies, dimensions and characteristic frequencies of the system. These so generated surrogates are shadowed by a trajectory of the system which starts at different initial conditions than the time series in question. They can be used then to test for complex synchronisation.
Angehende Physiklehrkräfte sollen im Rahmen ihres Studiums fachliches und fachdidaktisches Wissen erwerben, welches die Gestaltung lernförderlichen Unterrichts ermöglicht. Es ist allerdings empirisch nur wenig geklärt, wie sich dieses Wissen im Laufe des Studiums entwickelt und ob es zur Ausbildung von Handlungsfähigkeiten beiträgt. Um derartige Wirkungsaussagen treffen zu können, müssen Instrumente entwickelt werden, die eine valide Testwertinterpretation zulassen. In diesem Beitrag werden auf Basis von im Projekt Profile-P+ entwickelten Instrumenten Validitätsanalysen zur längsschnittlichen Entwicklung des Professionswissens von Physiklehramtsstudierenden im Verlauf des Bachelorstudiums und ihrer Fähigkeiten zur Planung und Reflexion von Physikunterricht sowie zum Erklären von physikalischen Sachverhalten vor und nach dem Praxissemester dargestellt. Neben Wissenstests kamen standardisierte Performanztests zum Einsatz. Die vorliegenden Ergebnisse sprechen dafür, dass die erhobenen Messwerte im Sinne von Wirkungsaussagen interpretiert werden können.
The occurrence of earthquakes is characterized by a high degree of spatiotemporal complexity. Although numerous patterns, e.g. fore- and aftershock sequences, are well-known, the underlying mechanisms are not observable and thus not understood. Because the recurrence times of large earthquakes are usually decades or centuries, the number of such events in corresponding data sets is too small to draw conclusions with reasonable statistical significance. Therefore, the present study combines both, numerical modeling and analysis of real data in order to unveil the relationships between physical mechanisms and observational quantities. The key hypothesis is the validity of the so-called "critical point concept" for earthquakes, which assumes large earthquakes to occur as phase transitions in a spatially extended many-particle system, similar to percolation models. New concepts are developed to detect critical states in simulated and in natural data sets. The results indicate that important features of seismicity like the frequency-size distribution and the temporal clustering of earthquakes depend on frictional and structural fault parameters. In particular, the degree of quenched spatial disorder (the "roughness") of a fault zone determines whether large earthquakes occur quasiperiodically or more clustered. This illustrates the power of numerical models in order to identify regions in parameter space, which are relevant for natural seismicity. The critical point concept is verified for both, synthetic and natural seismicity, in terms of a critical state which precedes a large earthquake: a gradual roughening of the (unobservable) stress field leads to a scale-free (observable) frequency-size distribution. Furthermore, the growth of the spatial correlation length and the acceleration of the seismic energy release prior to large events is found. The predictive power of these precursors is, however, limited. Instead of forecasting time, location, and magnitude of individual events, a contribution to a broad multiparameter approach is encouraging.