648 research outputs found
Fractal Dimensions in Classical and Quantum Mechanical Open Chaotic Systems
Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems.
In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time , infinite sample size , and infinite resolution (infinitesimal box-size ).
Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension . We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on , , and in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields.
In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension on and . We numerically explore this in weakly chaotic leaky systems.
There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum maps.Es ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet.
In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit , unendlicher Stichprobengröße und unendlicher Auflösung, also infinitesimaler Boxgröße auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden.
Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension durch und zeigen, dass die drei Parameter , und in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von , und anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht.
Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an:
Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen -Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen.
Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von von und auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist.
Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben
Layered Chaos in Mean-field and Quantum Many-body Dynamics
We investigate the dimension of the phase space attractor of a quantum
chaotic many-body ratchet in the mean-field limit. Specifically, we explore a
driven Bose-Einstein condensate in three distinct dynamical regimes - Rabi
oscillations, chaos, and self-trapping regime, and for each of them we
calculate the correlation dimension. For the ground state of the ratchet formed
by a system of field-free non-interacting particles, we find four distinct
pockets of chaotic dynamics throughout these regimes. We show that a
measurement of a local density in each of the dynamical regimes, has an
attractor characterized with a higher fractal dimension, ,
, and , as compared to the global measure
of current, , , and .
We find that the many-body case converges to mean-field limit with strong
sub-unity power laws in particle number , namely with
, and
for each of the dynamical regimes mentioned above.
The deviation between local and global measurement of the attractor's dimension
corresponds to an increase towards high condensate depletion which remains
constant for long time scales in both Rabi and chaotic regimes. The depletion
is found to scale polynomially with particle number as with
and for the two regimes.
Thus, we find a strong deviation from the mean-field results, especially in the
chaotic regime of the quantum ratchet. The ratchet also reveals quantum
revivals in the Rabi and self-trapped regimes but not in the chaotic regime.
Based on the obtained results we outline pathways for the identification and
characterization of the emergent phenomena in driven many-body systems
Nonlinear system identification
The prediction of a single observable time series has been achieved with reasonable accuracy and duration for the nonlinear systems developed by Rossler and Lorenz. Based on Takens\u27 Delay-vector Space, an artificial system has been generated using a polynomial least squares technique that includes all possible fifth order combinations of the vectors in the delay space. Furthermore, an optimum shift value has been shown to exist, such that any deviation decreases the accuracy and stability of the prediction. Additionally, an augmented form of the autocorrelation function, similar to the delay vector expansion, has been investigated. The first inflection of this correlation, typically in the dimension of the system, tends to coincide with the optimum shift value required for the best prediction. This method has also been utilized in conjunction with the Grassberger-Procaccia Distance correlation function to accurately determine the fractal dimension of the systems being investigated
Nonlinear time-series analysis of Hyperion's lightcurves
Hyperion is a satellite of Saturn that was predicted to remain in a chaotic
rotational state. This was confirmed to some extent by Voyager 2 and Cassini
series of images and some ground-based photometric observations. The aim of
this aticle is to explore conditions for potential observations to meet in
order to estimate a maximal Lyapunov Exponent (mLE), which being positive is an
indicator of chaos and allows to characterise it quantitatively. Lightcurves
existing in literature as well as numerical simulations are examined using
standard tools of theory of chaos. It is found that existing datasets are too
short and undersampled to detect a positive mLE, although its presence is not
rejected. Analysis of simulated lightcurves leads to an assertion that
observations from one site should be performed over a year-long period to
detect a positive mLE, if present, in a reliable way. Another approach would be
to use 2---3 telescopes spread over the world to have observations distributed
more uniformly. This may be achieved without disrupting other observational
projects being conducted. The necessity of time-series to be stationary is
highly stressed.Comment: 34 pages, 12 figures, 4 tables; v2 after referee report; matches the
version accepted in Astrophysics and Space Scienc
Analysis of deterministic chaotic signals
The prediction of a single observable time series has been achieved with varying degrees of success. The quality and duration of the prediction is dependent on many factors, the two most important being the reconstruction technique and the quantity of data. The goal of this work is to reduce the computational effort required to achieve satisfactory predictions. Without new methods, which are beyond the scope of this work, this requires a reduction in the size of the data set.
This thesis expands on earlier works using the delay vector space method and the autocorrelation function for reconstruction and applies this analysis technique to a well known non-linear dynamic system. The embedding delay and the sampling rate were varied while keeping the number of points the same in order to study the effects of varying the sampling rate. The results of this experimentation show the importance of the sampling rate and duration of the sample in the reconstruction and prediction. It is shown that the sampling duration may be more important than the number of points. It is apparent from this characteristic that a time series sampled over a longer duration may contain more information in fewer points
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