Iskanje po katalogu digitalne knjižnice

Opis: In this thesis the classical and quantum dynamics in billiard systems are considered. Extensive numerical studies of the classical transport properties in several examples of billiard families including the ergodic Bunimovich stadium and cut-circle billiards and the mixed-type Robnik and lemon billiards are performed. The analysis of the transport is based on the random model of diffusion which assumes that due the strongly chaotic dynamics the motion of the orbit on the discretized phase space is temporally uncorrelated. The cause of the deviations from the random model dynamics is traced to dynamical trapping due to stickiness. A novel approach to locally quantifying stickiness based on the statistics of the recurrence times is presented and applied to distinguish between exponential decays of recurrence times and other types of decays. This enables the identification of sticky areas in the chaotic components. Detailed maps of their structure for a wide range of parameter values, mapping the evolution of the mixed-phase spaces and revealing some particularly interesting special examples are presented. The recurrence time distributions in sticky areas are found to be well described by a mixture of exponential decays. The transport of particle ensembles in the momentum space of classical billiards is described by using an inhomogeneous diffusion model and the classical transport times are determined. The classical transport times are vital for the analysis of the localization of chaotic eigenstates in quantum billiards. The control parameter that describes the the degree of localization of the chaotic quantum eigenstates is the ratio between the Heisenberg time (Planck's constant divided by the mean level spacing) and the classical transport time. Extensive numerical calculations of the high-lying spectra and eigenstates of the stadium, Robnik and lemon quantum billiards are performed. The spectral statistics are analysed in terms of the standard methods of quantum chaos. The level repulsion exponent of localized eigenstates is found to be a rational function of the control parameter. The degree of localization is determined with respect to localization measures based on the Poincaré-Husimi representation of the eigenstates. The mean localization measure is found to be a rational function of the control parameter and linearly related to the level repulsion exponent. The distributions of the localization measures are analysed and found to be of a universal shape well described by a two parameter empirical distribution in billiards with no apparent stickiness. The nonuniversal system specific features of localization measure distributions are related to the presence of sticky areas in the phase spaces of classical billiards with specific examples shown.
Ključne besede: Transport, localization, chaos, quantum chaos, Hamiltonian systems, level spacing distribution, mixed phase space, billiard, quantum billiard, Husimi functions, stickiness, cantorus, chaotic eigenstates, level repulsion.
Objavljeno v DKUM: 13.01.2021; Ogledov: 999; Prenosov: 131
Celotno besedilo (24,93 MB)

Opis: In the present paper, we investigate the blast induced ground motion recorded at the limestone quarry “Suva Vrela” near Kosjerić, which is located in the western part of Serbia. We examine the recorded signals by means of surrogate data methods and a determinism test, in order to determine whether the recorded ground velocity is stochastic or deterministic in nature. Longitudinal, transversal and the vertical ground motion component are analyzed at three monitoring points that are located at different distances from the blasting source. The analysis reveals that the recordings belong to a class of stationary linear stochastic processes with Gaussian inputs, which could be distorted by a monotonic, instantaneous, time-independent nonlinear function. Low determinism factors obtained with the determinism test further confirm the stochastic nature of the recordings. Guided by the outcome of time series analysis, we propose an improved prediction model for the peak particle velocity based on a neural network. We show that, while conventional predictors fail to provide acceptable prediction accuracy, the neural network model with four main blast parameters as input, namely total charge, maximum charge per delay, distance from the blasting source to the measuring point, and hole depth, delivers significantly more accurate predictions that may be applicable on site. We also perform a sensitivity analysis, which reveals that the distance from the blasting source has the strongest influence on the final value of the peak particle velocity. This is in full agreement with previous observations and theory, thus additionally validating our methodology and main conclusions.
Ključne besede: blasting, vibrations, surrogate data, deterministic chaos, stochasticity
Objavljeno v DKUM: 19.06.2017; Ogledov: 759; Prenosov: 316
Celotno besedilo (1,45 MB)
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Ključne besede: cobweb, hiperincursivity, system dynamics, anticipative system, nonlinear system, Farey tree, chaos
Objavljeno v DKUM: 10.07.2015; Ogledov: 834; Prenosov: 34
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Ključne besede: hrup, intenziteta hrupa, dilema zapornika, kaos, nelinearni dinamični sistemi, noise, spatiotemporal noise, intensity, prisoner's dilemma, chaos, nonlinear dynamic systems
Objavljeno v DKUM: 10.07.2015; Ogledov: 1282; Prenosov: 27
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Opis: In the dissertation I have dealt with time-dependent (nonautonomous) systems, the conservative (Hamiltonian) as well as dissipative, and investigated their dynamical and statistical properties. In conservative (Hamiltonian) time-dependent systems the energy is not conserved, whilst the Liouville theorem about the conservation of the phase space volume still applies. We are interested to know, whether the system can gain energy, and whether this energy can grow unbounded, up to infinity, and we are interested in the system's behaviour in the mean, as well as its statistical properties. An example of such a system goes back to the 1940s, when Fermi proposed the acceleration of cosmic rays (in the first place protons) upon the collisions with moving magnetic domains in the interstellar medium of our Galaxy, and in other galaxies. He then proposed a simple mechanical one-dimensional model, the so-called Fermi-Ulam Model (FUM), where a point particle is moving between two rigid walls, one being at rest and the other one oscillating. If the oscillation is periodic and smooth, it turned out in a nontrivial way, which is, in the modern era of understanding the chaotic dynamical systems, well understood, namely that the unbounded increasing of the energy (the so-called Fermi acceleration) is not possible, due to the barriers in form of invariant tori, which partition the phase space into regions, between which the transitions are not possible. The research has then been extended to other simple dyanamical systems, which have complex dynamics. The first was so-called bouncer model, in which a point particle bounces off the oscillating platform in a gravitational field. In this simple system the Fermi acceleration is possible. Later the research was directed towards two-dimensional billiard systems. It turned out that the Fermi acceleration is possible in all such systems, which are at least partially chaotic (of the mixed type), or even in a system that is integrable as static, namely in case of the elliptic billiard. (The circle billiard is an exception, because it is always integrable, as the angular momentum is conserved even in time-dependent case.) The study of time-dependent systems has developed strongly worldwide around the 1990s, in particular in 2000s, and became one of the central topics in nonlinear dynamics. It turned out, quite generally, but formal and implicit, in the sense of mathematical existence theorems, that in nonautonomous Hamilton systems the energy can grow unbounded, meaning that the system ``pumps" the energy from the environment with which it interacts. There are many open questions: how does the energy increase with time, in particular in the mean of some representative ensemble of initial conditions (typically the phase space of two-dimensional time-dependent billiards is four-dimensional.) It turned out that almost everywhere the power laws apply, empirically, based on the numerical calculations, but with various acceleration exponents. If the Fermi acceleration is not posssible, like e.g. in the FUM, due to the invariant tori, then after a certain time of acceleration stage the crossover into the regime of saturation takes place, whose characteristics also follow the power laws. One of the central themes in the dissertation is the study of these power laws, their critical exponents, analytical relationships among them, using the scaling analysis (Leonel, McClintock and Silva, Phys. Rev. Lett. 2004). Furthermore, the central theme is the question, what happens, if, in a nonautonomous Hamilton system which exhibits Fermi acceleration, we introduce dissipation, either at the collisions with the walls (collisional dissipation) or during the free motion (in-flight dissipation, due to the viscosity of the fluid or the drag force etc.). Dissipation typically transforms the periodic points into point attractors and chaotic components into chaotic attractors. The Fermi acceleration is always suppressed. We are interested in the phase portraits of
Ključne besede: nonlinear dynamics, dynamical systems, conservative and dissipative systems, time-dependent systems, Fermi acceleration, billiards, kicked systems, chaos, chaotic and periodic attractors, bifurcations, boundary crisis
Objavljeno v DKUM: 19.09.2012; Ogledov: 2807; Prenosov: 135
Celotno besedilo (16,09 MB)