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Contents: 1 Introduction 2 Experiment 3 Data 4 Symbolic dynamics 4.1 Symbolic dynamics as a tool for data analysis 4.2 2-symbols coding 4.3 3-symbols coding 5 Measures of complexity 5.1 Word statistics 5.2 Shannon entropy 6 Testing for stationarity 6.1 Stationarity 6.2 Time series of cycle durations 6.3 Chi-square test 7 Control parameters in the production of rhythms 8 Analysis of relative phases 9 Discussion 10 Outlook
A numerical bifurcation analysis of the electrically driven plane sheet pinch is presented. The electrical conductivity varies across the sheet such as to allow instability of the quiescent basic state at some critical Hartmann number. The most unstable perturbation is the two-dimensional tearing mode. Restricting the whole problem to two spatial dimensions, this mode is followed up to a time-asymptotic steady state, which proves to be sensitive to three-dimensional perturbations even close to the point where the primary instability sets in. A comprehensive three-dimensional stability analysis of the two-dimensional steady tearing-mode state is performed by varying parameters of the sheet pinch. The instability with respect to three-dimensional perturbations is suppressed by a sufficiently strong magnetic field in the invariant direction of the equilibrium. For a special choice of the system parameters, the unstably perturbed state is followed up in its nonlinear evolution and is found to approach a three-dimensional steady state.
We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven fluid layer with stress-free top and bottom boundaries, a square horizontal periodicity with a small aspect ratio, and rotation at low to moderate rates about a vertical axis. The Prandtl number is 6.8. If the rotation is not too slow, the skewed-varicose instability leads from stationary rolls to a stationary mixed-mode solution, which in turn loses stability to a heteroclinic cycle formed by unstable roll states and connections between them. The unstable eigenvectors of these roll states are also of the skewed-varicose or mixed-mode type and in some parameter regions skewed-varicose like shearing oscillations as well as square patterns are involved in the cycle. Always present weak noise leads to irregular horizontal translations of the convection pattern and makes the dynamics chaotic, which is verified by calculating Lyapunov exponents. In the nonrotating case, the primary rolls lose, depending on the aspect ratio, stability to traveling waves or a stationary square pattern. We also study the symmetries of the solutions at the intermittent fixed points in the heteroclinic cycle.
Our dynamic Sun manifests its activity by different phenomena: from the 11-year cyclic sunspot pattern to the unpredictable and violent explosions in the case of solar flares. During flares, a huge amount of the stored magnetic energy is suddenly released and a substantial part of this energy is carried by the energetic electrons, considered to be the source of the nonthermal radio and X-ray radiation. One of the most important and still open question in solar physics is how the electrons are accelerated up to high energies within (the observed in the radio emission) short time scales. Because the acceleration site is extremely small in spatial extent as well (compared to the solar radius), the electron acceleration is regarded as a local process. The search for localized wave structures in the solar corona that are able to accelerate electrons together with the theoretical and numerical description of the conditions and requirements for this process, is the aim of the dissertation. Two models of electron acceleration in the solar corona are proposed in the dissertation: I. Electron acceleration due to the solar jet interaction with the background coronal plasma (the jet--plasma interaction) A jet is formed when the newly reconnected and highly curved magnetic field lines are relaxed by shooting plasma away from the reconnection site. Such jets, as observed in soft X-rays with the Yohkoh satellite, are spatially and temporally associated with beams of nonthermal electrons (in terms of the so-called type III metric radio bursts) propagating through the corona. A model that attempts to give an explanation for such observational facts is developed here. Initially, the interaction of such jets with the background plasma leads to an (ion-acoustic) instability associated with growing of electrostatic fluctuations in time for certain range of the jet initial velocity. During this process, any test electron that happen to feel this electrostatic wave field is drawn to co-move with the wave, gaining energy from it. When the jet speed has a value greater or lower than the one, required by the instability range, such wave excitation cannot be sustained and the process of electron energization (acceleration and/or heating) ceases. Hence, the electrons can propagate further in the corona and be detected as type III radio burst, for example. II. Electron acceleration due to attached whistler waves in the upstream region of coronal shocks (the electron--whistler--shock interaction) Coronal shocks are also able to accelerate electrons, as observed by the so-called type II metric radio bursts (the radio signature of a shock wave in the corona). From in-situ observations in space, e.g., at shocks related to co-rotating interaction regions, it is known that nonthermal electrons are produced preferably at shocks with attached whistler wave packets in their upstream regions. Motivated by these observations and assuming that the physical processes at shocks are the same in the corona as in the interplanetary medium, a new model of electron acceleration at coronal shocks is presented in the dissertation, where the electrons are accelerated by their interaction with such whistlers. The protons inflowing toward the shock are reflected there by nearly conserving their magnetic moment, so that they get a substantial velocity gain in the case of a quasi-perpendicular shock geometry, i.e, the angle between the shock normal and the upstream magnetic field is in the range 50--80 degrees. The so-accelerated protons are able to excite whistler waves in a certain frequency range in the upstream region. When these whistlers (comprising the localized wave structure in this case) are formed, only the incoming electrons are now able to interact resonantly with them. But only a part of these electrons fulfill the the electron--whistler wave resonance condition. Due to such resonant interaction (i.e., of these electrons with the whistlers), the electrons are accelerated in the electric and magnetic wave field within just several whistler periods. While gaining energy from the whistler wave field, the electrons reach the shock front and, subsequently, a major part of them are reflected back into the upstream region, since the shock accompanied with a jump of the magnetic field acts as a magnetic mirror. Co-moving with the whistlers now, the reflected electrons are out of resonance and hence can propagate undisturbed into the far upstream region, where they are detected in terms of type II metric radio bursts. In summary, the kinetic energy of protons is transfered into electrons by the action of localized wave structures in both cases, i.e., at jets outflowing from the magnetic reconnection site and at shock waves in the corona.
The solar tachocline is a thin transition layer between the solar radiative zone rotating uniformly and the solar convection zone, which has a mainly latitudinal differential rotation profile. This layer has a thickness of less than $0.05R_{\sun}$ and is subject to extreme radial as well as latitudinal shears. Helioseismological estimates put this layer at roughly $0.7R_{\sun}$. The tachocline mostly resides in the sub-adiabatic, non-turbulent radiative interior, except for a small overlap with the convection zone on the top. Many proposed dynamo mechanisms involve strong toroidal magnetic fields in this transition region. The exact mechanisms behind the formation of such a thin layer is still disputed. A very plausible mechanism is the one involving a weak, relic poloidal magnetic field trapped inside the radiative zone, which is responsible for expelling differential rotation outwards. This was first proposed by \citet{RK97}. The present work develops this idea with numerical simulations including additional effects like meridional circulation. It is shown that a relic field of 1~Gauss or smaller would be sufficient to explain the observed thickness of the tachocline. The stability of the solar tachocline is addressed as the next part of the problem. It is shown that the tachocline is stable up to a differential rotation of 52\% in the absence of magnetic fields. This is a new finding as compared to the earlier two dimensional models which estimated the solar differential rotation (about 28\%) to be marginally stable or even unstable. The changed stability limit is attributed to the changed stability criterion of the 3-dimensional model which also involves radial gradients of the angular velocity. In the presence of toroidal magnetic field belts, the lowest non-axisymmetric mode is shown to be the most unstable one for the radiative part of the tachocline. It is estimated that the tachocline would become unstable for toroidal fields exceeding about 100~Gauss. With both formation and stability questions satisfactorily addressed, this work presents the most comprehensive analysis of the physical processes in the solar tachocline to date.
Our Solar system contains a large amount of dust, containing valuable information about our close cosmic environment. If created in a planet's system, the particles stay predominantly in its vicinity and can form extended dust envelopes, tori or rings around them. A fascinating example of these complexes are Saturnian rings containing a wide range of particles sizes from house-size objects in the main rings up to micron-sized grains constituting the E ring. Other example are ring systems in general, containing a large fraction of dust or also the putative dust-tori surrounding the planet Mars. The dynamical life'' of such circumplanetary dust populations is the main subject of our study. In this thesis a general model of creation, dynamics and death'' of circumplanetary dust is developed. Endogenic and exogenic processes creating dust at atmosphereless bodies are presented. Then, we describe the main forces influencing the particle dynamics and study dynamical responses induced by stochastic fluctuations. In order to estimate the properties of steady-state population of considered dust complex, the grain mean lifetime as a result of a balance of dust creation, life'' and loss mechanisms is determined. The latter strongly depends on the surrounding environment, the particle properties and its dynamical history. The presented model can be readily applied to study any circumplanetary dust complex. As an example we study dynamics of two dust populations in the Solar system. First we explore the dynamics of particles, ejected from Martian moon Deimos by impacts of micrometeoroids, which should form a putative tori along the orbit of the moon. The long-term influence of indirect component of radiation pressure, the Poynting-Robertson drag gives rise in significant change of torus geometry. Furthermore, the action of radiation pressure on rotating non-spherical dust particles results in stochastic dispersion of initially confined ensemble of particles, which causes decrease of particle number densities and corresponding optical depth of the torus. Second, we investigate the dust dynamics in the vicinity of Saturnian moon Enceladus. During three flybys of the Cassini spacecraft with Enceladus, the on-board dust detector registered a micron-sized dust population around the moon. Surprisingly, the peak of the measured impact rate occurred 1 minute before the closest approach of the spacecraft to the moon. This asymmetry of the measured rate can be associated with locally enhanced dust production near Enceladus south pole. Other Cassini instruments also detected evidence of geophysical activity in the south polar region of the moon: high surface temperature and extended plumes of gas and dust leaving the surface. Comparison of our results with this in situ measurements reveals that the south polar ejecta may provide the dominant source of particles sustaining the Saturn's E ring.
In nature one commonly finds interacting complex oscillators which by the coupling scheme form small and large networks, e.g. neural networks. Surprisingly, the oscillators can synchronize, still preserving the complex behavior. Synchronization is a fundamental phenomenon in coupled nonlinear oscillators. Synchronization can be enhanced at different levels, that is, the constraints on which the synchronization appears. Those can be in the trajectory amplitude, requiring the amplitudes of both oscillators to be equal, giving place to complete synchronization. Conversely, the constraint could also be in a function of the trajectory, e.g. the phase, giving place to phase synchronization (PS). In this case, one requires the phase difference between both oscillators to be finite for all times, while the trajectory amplitude may be uncorrelated. The study of PS has shown its relevance to important technological problems, e.g. communication, collective behavior in neural networks, pattern formation, Parkinson disease, epilepsy, as well as behavioral activities. It has been reported that it mediates processes of information transmission and collective behavior in neural and active networks and communication processes in the Human brain. In this work, we have pursed a general way to analyze the onset of PS in small and large networks. Firstly, we have analyzed many phase coordinates for compact attractors. We have shown that for a broad class of attractors the PS phenomenon is invariant under the phase definition. Our method enables to state about the existence of phase synchronization in coupled chaotic oscillators without having to measure the phase. This is done by observing the oscillators at special times, and analyzing whether this set of points is localized. We have show that this approach is fruitful to analyze the onset of phase synchronization in chaotic attractors whose phases are not well defined, as well as, in networks of non-identical spiking/bursting neurons connected by chemical synapses. Moreover, we have also related the synchronization and the information transmission through the conditional observations. In particular, we have found that inside a network clusters may appear. These can be used to transmit more than one information, which provides a multi-processing of information. Furthermore, These clusters provide a multichannel communication, that is, one can integrate a large number of neurons into a single communication system, and information can arrive simultaneously at different places of the network.
Ziel dieser Arbeit ist die phänomenologische Untersuchung der Feuchteempfindlichkeit der elektrischen Eigenschaften dünner Polymerschichten. Diese Untersuchungen stellen gleichzeitig Vorarbeiten zur Entwicklung von Prototypen von zwei polymeren Dünnschicht-Feuchtesensoren dar, die sich durch die spezielle Auswahl der feuchtesensitiven Materialien jeweils durch eine besondere Eigenschaft gegenüber kommerziellen Massenprodukten auszeichnen. Ziel der Entwicklungsarbeiten für den ersten Prototypen war die Konstruktion eines schnellen Feuchtesensors, der plötzliche und sprunghafte Feuchteänderungen in der umgebenden Atmosphäre möglichst rasch detektieren kann. Dafür wurden dünne Schichten von Poly-DADMAC auf Interdigitalstrukturen aufgebracht, die einen möglichst direkten Kontakt zwischen feuchtesensitiver Schicht und umgebender, feuchter Atmosphäre gewährleisten. Als Messgrößen dienten die Wechselstromgrößen Widerstand und Kapazität der Schichten. Die Feuchtekennlinien der Schichten zeigen gute Konstanz und hohe Reproduzierbarkeit. Der Widerstand der Schichten ändert sich durch den Einfluss von Feuchte je nach Schichtdicke um 3 bis 5 Größenordnungen und eignet sich als Messgröße für die Feuchtigkeit im gesamten Feuchtebereich. Die Hysterese der Filme konnte auf kleiner als 2,5% r.F. bestimmt werden, die Reproduzierbarkeit auf besser als 1% r.F. Die Ansprechzeit der Schichten lässt sich schichtdickenabhängig zu 1 bis 10 Sekunden bestimmen. Hierbei zeigen besonders die dünnen Schichten kurze Ansprechzeiten. Zielstellung für den zweiten Feuchtesensor war die Entwicklung eines Prototypen, dessen sensitive Schicht sich biostatisch und biozid verhält, so dass er in biotischen Umgebungen eingesetzt werden kann. Es wurden fünf Polysulfobetaine synthetisiert, deren Biozidität und Biostatik mit dem Kontakttest nach Rönnpagel, dem ISO846-Test und Abbautests bestimmt wurde. Zwei Polymere – Poly-DMMAAPS (BT2) und Poly-[MSA-Styren-Sulfobetain] (BT5) – erwiesen sich als ausreichend biozid und biostatisch. Schichten dieser Polymere wurden auf Interdigitalstrukturen aufgezogen, anschließend wurden die Kennlinien dieser Proben aufgenommen. Die Messwerte zeigen für beide Polymere gute Konstanz und eine hohe Reproduzierbarkeit. BT2-Proben sind zwischen 20% und 80% r.F. besonders empfindlich und zeigen über einen Monat keine Langzeitdrift. Vernetzte Proben zeigen bis 50°C keinen temperaturbedingten Abfall der Feuchteempfindlichkeit. Der Einsatz vernetzter BT5-Schichten als kapazitiver Feuchtesensor ist bis etwa 70°C möglich, die Schichten sind selbst nach Lagerung im Hochvakuum und mehrfacher Betauung stabil. Damit liegen zwei funktionsfähige Prototypen von Feuchtesensoren vor, für die die meisten Kennwerte denen von vergleichbaren kommerziellen Feuchtesensoren entsprechen. Gleichzeitig zeichnen sie sich aber durch eine sehr niedrige Ansprechzeit bzw. eine ausreichende Lebensdauer unter biotischen Bedingungen aus.
Box-Simulationen von rotierender Magnetokonvektion im flüssigen Erdkern Numerische Simulationen der 3D-MHD Gleichungen sind mit Hilfe des Codes NIRVANA durchgeführt worden. Die Gleichungen für kompressible rotierende Magnetokonvektion wurden für erdähnliche Bedingungen numerisch in einer kartesischen Box gelöst. Charakteristische Eigenschaften mittlerer Größen, wie der Turbulenz-Intensität oder der turbulente Wärmefluss, die durch die kombinierte Wirkung kleinskaliger Fluktuationen entstehen, wurden bestimmt. Die Korrelationslänge der Turbulenz hängt signifikant von der Stärke und der Orientierung des Magnetfeldes ab, und das anisotrope Verhalten der Turbulenz aufgrund von Coriolis- und Lorentzkraft ist für schnellere Rotation wesentlich stärker ausgeprägt. Die Ausbildung eines isotropen Verhaltens auf kleinen Skalen unter dem Einfluss von Rotation alleine wird bereits durch ein schwaches Magnetfeld verhindert. Dies resultiert in einer turbulenten Strömung, die durch die vertikale Komponente dominiert wird. In Gegenwart eines horizontalen Magnetfeldes nimmt der vertikale turbulente Wärmefluss leicht mit zunehmender Feldstärke zu, so dass die Kühlung eines rotierenden Systems verbessert wird. Der horizontale Wärmetransport ist stets westwärts und in Richtung der Pole orientiert. Letzteres kann unter Umständen die Quelle für eine großskalige meridionale Strömung darstellen, während erstes in globalen Simulationen mit nicht axialsymmetrischen Randbedingungen für den Wärmefluss von Bedeutung ist. Die mittlere elektromotorische Kraft, die die Erzeugung von magnetischem Fluss durch die Turbulenz beschreibt, wurde unmittelbar aus den Lösungen für Geschwindigkeit und Magnetfeld berechnet. Hieraus konnten die entsprechenden α-Koeffizienten hergeleitet werden. Aufgrund der sehr schwachen Dichtestratifizierung ändert der α-Effekt sein Vorzeichen nahezu exakt in der Mitte der Box. Der α-Effekt ist positiv in der oberen Hälfte und negativ in der unteren Hälfte einer auf der Nordhalbkugel rotierenden Box. Für ein starkes Magnetfeld ergibt sich zudem eine deutliche abwärts orientierte Advektion von magnetischem Fluss. Ein Mean-Field Modell des Geodynamos wurde konstruiert, das auf dem α-Effekt basiert, wie er aus den Box-Simulationen berechnet wurde. Für eine äußerst beschränkte Klasse von radialen α-Profilen weist das lineare α^2-Modell Oszillationen auf einer Zeitskala auf, die durch die turbulente Diffusionszeit bestimmt wird. Die wesentlichen Eigenschaften der periodischen Lösungen werden präsentiert, und der Einfluss der Größe des inneren Kerns auf die Charakteristiken des kritischen Bereichs, innerhalb dessen oszillierende Lösungen auftreten, wurden untersucht. Reversals werden als eine halbe Oszillation interpretiert. Sie sind ein recht seltenes Ereignis, da sie lediglich dann stattfinden können, wenn das α-Profil ausreichend lange in dem periodische Lösungen erlaubenden Bereich liegt. Aufgrund starker Fluktuationen auf der konvektiven Zeitskala ist die Wahrscheinlichkeit eines solchen Reversals relativ klein. In einem einfachen nicht-linearen Mean-Field Modell mit realistischen Eingabeparametern, die auf den Box-Simulationen beruhen, konnte die Plausibilität des Reversal-Modells anhand von Langzeitsimulationen belegt werden.
It is shown that the ff effect of mean-field magnetohydrodynamics, which consists in the generation of a mean electromotive force along the mean magnetic field by turbulently fluctuating parts of velocity and magnetic field, is equivalent to the simultaneous generation of both turbulent and mean-field magnetic helicities, the generation rates being equal in magnitude and opposite in sign. In the particular case of statistically stationary and homogeneous fluctuations this implies that the ff effect can increase the energy in the mean magnetic field only under the condition that also magnetic helicity is accumulated there.
We report on bifurcation studies for the incompressible Navier-Stokes equations in two space dimensions with periodic boundary conditions and an external forcing of the Kolmogorov type. Fourier representations of velocity and pressure have been used to approximate the original partial differential equations by a finite-dimensional system of ordinary differential equations, which then has been studied by means of bifurcation-analysis techniques. A special route into chaos observed for increasing Reynolds number or strength of the imposed forcing is described. It includes several steady states, traveling waves, modulated traveling waves, periodic and torus solutions, as well as a period-doubling cascade for a torus solution. Lyapunov exponents and Kaplan-Yorke dimensions have been calculated to characterize the chaotic branch. While studying the dynamics of the system in Fourier space, we also have transformed solutions to real space and examined the relation between the different bifurcations in Fourier space and toplogical changes of the streamline portrait. In particular, the time-dependent solutions, such as, e.g., traveling waves, torus, and chaotic solutions, have been characterized by the associated fluid-particle motion (Lagrangian dynamics).
Projection methods based on wavelet functions combine optimal convergence rates with algorithmic efficiency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguelette decompositions enable the realization of especially fast algorithms for certain operators.
The bifurcation behaviour of the 3D magnetohydrodynamic equations has been studied for external forcings of varying degree of helicity. With increasing strength of the forcing a primary non-magnetic steady state loses stability to a magnetic periodic state if the helicity exceeds a threshold value and to different non-magnetic states otherwise.
The present paper is related to the problem of approximating the exact solution to the magnetohydrodynamic equations (MHD). The behaviour of a viscous, incompressible and resistive fluid is exemined for a long period of time. Contents: 1 The magnetohydrodynamic equations 2 Notations and precise functional setting of the problem 3 Existence, uniqueness and regularity results 4 Statement and Proof of the main theorem 5 The approximate inertial manifold 6 Summary
We demonstrate the occurrence of regimes with singular continuous (fractal) Fourier spectra in autonomous dissipative dynamical systems. The particular example in an ODE system at the accumulation points of bifurcation sequences associated to the creation of complicated homoclinic orbits. Two different machanisms responsible for the appearance of such spectra are proposed. In the first case when the geometry of the attractor is symbolically represented by the Thue-Morse sequence, both the continuous-time process and its descrete Poincaré map have singular power spectra. The other mechanism owes to the logarithmic divergence of the first return times near the saddle point; here the Poincaré map possesses the discrete spectrum, while the continuous-time process displays the singular one. A method is presented for computing the multifractal characteristics of the singular continuous spectra with the help of the usual Fourier analysis technique.
The Voyager 2 Photopolarimeter experiment has yielded the highest resolved data of Saturn's rings, exhibiting a wide variety of features. The B-ring region between 105000 km and 110000 km distance from Saturn has been investigated. It has a high matter density and contains no significance features visible by eye. Analysis with statistical methods has let us to the detection of two significant events. These features are correlated with the inner 3:2 resonances of the F-ring shepherd satellites Pandora and Prometheus, and may be evidence of large ring paricles caught in the corotation resonances.
We report on bifurcation studies for the incompressible magnetohydrodynamic equations in three space dimensions with periodic boundary conditions and a temporally constant external forcing. Fourier reprsentations of velocity, pressure and magnetic field have been used to transform the original partial differential equations into systems of ordinary differential equations (ODE), to which then special numerical methods for the qualitative analysis of systems of ODE have been applied, supplemented by the simulative calculation of solutions for selected initial conditions. In a part of the calculations, in order to reduce the number of modes to be retained, the concept of approximate inertial manifolds has been applied. For varying (incereasing from zero) strength of the imposed forcing, or varying Reynolds number, respectively, time-asymptotic states, notably stable stationary solutions, have been traced. A primary non-magnetic steady state loses, in a Hopf bifurcation, stability to a periodic state with a non-vanishing magnetic field, showing the appearance of a generic dynamo effect. From now on the magnetic field is present for all values of the forcing. The Hopf bifurcation is followed by furhter, symmetry-breaking, bifurcations, leading finally to chaos. We pay particular attention to kinetic and magnetic helicities. The dynamo effect is observed only if the forcing is chosen such that a mean kinetic helicity is generated; otherwise the magnetic field diffuses away, and the time-asymptotic states are non-magnetic, in accordance with traditional kinematic dynamo theory.
It is desirable to reduce the potential threats that result from the variability of nature, such as droughts or heat waves that lead to food shortage, or the other extreme, floods that lead to severe damage. To prevent such catastrophic events, it is necessary to understand, and to be capable of characterising, nature's variability. Typically one aims to describe the underlying dynamics of geophysical records with differential equations. There are, however, situations where this does not support the objectives, or is not feasible, e.g., when little is known about the system, or it is too complex for the model parameters to be identified. In such situations it is beneficial to regard certain influences as random, and describe them with stochastic processes. In this thesis I focus on such a description with linear stochastic processes of the FARIMA type and concentrate on the detection of long-range dependence. Long-range dependent processes show an algebraic (i.e. slow) decay of the autocorrelation function. Detection of the latter is important with respect to, e.g. trend tests and uncertainty analysis. Aiming to provide a reliable and powerful strategy for the detection of long-range dependence, I suggest a way of addressing the problem which is somewhat different from standard approaches. Commonly used methods are based either on investigating the asymptotic behaviour (e.g., log-periodogram regression), or on finding a suitable potentially long-range dependent model (e.g., FARIMA[p,d,q]) and test the fractional difference parameter d for compatibility with zero. Here, I suggest to rephrase the problem as a model selection task, i.e.comparing the most suitable long-range dependent and the most suitable short-range dependent model. Approaching the task this way requires a) a suitable class of long-range and short-range dependent models along with suitable means for parameter estimation and b) a reliable model selection strategy, capable of discriminating also non-nested models. With the flexible FARIMA model class together with the Whittle estimator the first requirement is fulfilled. Standard model selection strategies, e.g., the likelihood-ratio test, is for a comparison of non-nested models frequently not powerful enough. Thus, I suggest to extend this strategy with a simulation based model selection approach suitable for such a direct comparison. The approach follows the procedure of a statistical test, with the likelihood-ratio as the test statistic. Its distribution is obtained via simulations using the two models under consideration. For two simple models and different parameter values, I investigate the reliability of p-value and power estimates obtained from the simulated distributions. The result turned out to be dependent on the model parameters. However, in many cases the estimates allow an adequate model selection to be established. An important feature of this approach is that it immediately reveals the ability or inability to discriminate between the two models under consideration. Two applications, a trend detection problem in temperature records and an uncertainty analysis for flood return level estimation, accentuate the importance of having reliable methods at hand for the detection of long-range dependence. In the case of trend detection, falsely concluding long-range dependence implies an underestimation of a trend and possibly leads to a delay of measures needed to take in order to counteract the trend. Ignoring long-range dependence, although present, leads to an underestimation of confidence intervals and thus to an unjustified belief in safety, as it is the case for the return level uncertainty analysis. A reliable detection of long-range dependence is thus highly relevant in practical applications. Examples related to extreme value analysis are not limited to hydrological applications. The increased uncertainty of return level estimates is a potentially problem for all records from autocorrelated processes, an interesting examples in this respect is the assessment of the maximum strength of wind gusts, which is important for designing wind turbines. The detection of long-range dependence is also a relevant problem in the exploration of financial market volatility. With rephrasing the detection problem as a model selection task and suggesting refined methods for model comparison, this thesis contributes to the discussion on and development of methods for the detection of long-range dependence.
In the modern industrialized countries every year several hundred thousands of people die due to the sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, non-invasive diagnostic tools like Holter-monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyse the HRV. Especially, some complexity measures that are basing on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients.
We have used techniques of nonlinear dynamics to compare a special model for the reversals of the Earth's magnetic field with the observational data. Although this model is rather simple, there is no essential difference to the data by means of well-known characteristics, such as correlation function and probability distribution. Applying methods of symbolic dynamics we have found that the considered model is not able to describe the dynamical properties of the observed process. These significant differences are expressed by algorithmic complexity and Renyi information.