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Dissertations / Theses on the topic 'Dynamical chaos'
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Krcelic, Khristine M. "Chaos and Dynamical Systems." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1364545282.
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Bird, C. M. "The control of chaos." Thesis, University of Surrey, 1996. http://epubs.surrey.ac.uk/804952/.
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Chygryn, S. A. "Intermittent chaos in Hamiltonian dynamical systems." Thesis, Сумський державний університет, 2014. http://essuir.sumdu.edu.ua/handle/123456789/35109.
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The statistical characterization of chaotic trajectories in Hamiltonian dynamical systems attract special interest. Such systems usually show coexistence of regions of chaotic and regular motion in the phase space. When chaotic trajectories approach the regular regions, they stick to their border inducing long periods of almost regular motion. This intermittent behavior determines the main dynamical properties of the system. The fundamental problem is how to quantitatively relate the intermittency of the chaotic dynamics to the distribution and stability properties of the regular regions of the phase space.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/35109
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Ruiter, Julia. "Practical Chaos: Using Dynamical Systems to Encrypt Audio and Visual Data." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/scripps_theses/1389.
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Although dynamical systems have a multitude of classical uses in physics and applied mathematics, new research in theoretical computer science shows that dynamical systems can also be used as a highly secure method of encrypting data. Properties of Lorenz and similar systems of equations yield chaotic outputs that are good at masking the underlying data both physically and mathematically. This paper aims to show how Lorenz systems may be used to encrypt text and image data, as well as provide a framework for how physical mechanisms may be built using these properties to transmit encrypted wave signals.
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Orrell, D. J. "Modelling nonlinear dynamical systems : chaos, uncertainty and error." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.393997.
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Wilson, Howard B. "Applications of dynamical systems in ecology." Thesis, University of Warwick, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.387403.
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Beale, L. C. "Chaos, strange attractors and bifurcations in dissipative dynamical systems." Thesis, University of Canterbury. Mathematics, 1988. http://hdl.handle.net/10092/8410.
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In this dissertation a study is made of chaotic behaviour, the bifurcation sequences leading to chaos and the manifestation of chaos in the form of a strange attractor in dissipative dynamical systems.
In chapter 1 we provide an overview of the material covered in this review and introduce several concepts from the basic theory of dynamical systems, such as Poincaré return maps and simple bifurcations.
After introducing the concept of chaos and strange attractors in dissipative dynamical systems, we divide higher dimensional systems into three categories in chapter 2. Each is illustrated with examples. Central to the discussion is the well studied Lorenz system. Other important mathematical models are looked at, in particular the Rössler model and the two-dimensional Hénon map.
The various measures of dimension, in the fractal context, and the numerical methods currently in use for determining these quantities are presented in chapter 3.
In view of their relative computational simplicity and direct relevance to chaos, one-dimensional mappings are looked at in chapter 4.
In chapter 5, the idea of the transition to turbulence being a chaotic regime is introduced and the various routes to turbulence are examined in turn.
In chapter 6, we present a Fourier series method for approximating the phase-space trajectories of a dynamical system. We illustrate the technique by carrying out the calculations required on the equations describing the evolution of the spherical pendulum model of Miles (l984b).
No attempt is made to cover the whole field of research chaos.
The use of symbolic dynamics is avoided wherever possible for simplicity and brevity in this review.
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Siegwart, David Kevin. "Classical and quantum chaos of dynamical systems : rotating billiards." Thesis, Durham University, 1990. http://etheses.dur.ac.uk/6228/.
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The theory of classical chaos is reviewed. From the definition of integrable systems using the Hamilton-Jacobi equation, the theory of perturbed systems is developed and the Kolmogorov-Arnold-Moser (KAM) theorem is explained. It is shown how chaotic motion in Hamiltonian systems is governed by the in tricate connections of stable and unstable invariant manifolds, and how it can be catagorised by algorithmic complexity and symbolic dynamics, giving chaotic measures such as Lyapunov exponents and Kolmogorov entropy. Also reviewed is Gutzwiller's semiclassical trace formula for strongly chaotic systems, torus quantisation for integrable systems, the asymptotic level density for stationary billiards, and random matrix theories for describing spectral fluctuation properties. The classical theory is applied to rotating billiards, particularly the free motion of a particle in a circular billiard rotating uniformly in its own plane about a point on its edge. Numerically, it is shown that the system's classical behaviour ranges from fully chaotic at intermediate energies, to completely integrable at very low and very high energies. It is shown that the strong chaos is due to discontinuities in the Poincare map, caused by trajectories which just glance the boundary-an effect of the curvature of trajectories. Weaker chaos exists due to the usual folding and stretching of the Hamiltonian flow. Approximate invariant curves for integrable motion are found, valid far from the presence of glancing trajectories. The major structures of phase space are investigated: a fixed point and its bifurcation into a two-cycle, and their stabilities. Lyapunov exponents for trajectories are calculated and the chaotic volume for a wide range of energies is measured. Quantum mechanically, the energy spectrum of the system is found numerically. It is shown that at the energies where the classical system is completely integrable the levels do not repel, and at those energies where it is completely chaotic there is strong level repulsion. The nearest neighbour level spacing distributions for various ranges of energy and values of Planck's constant are found. In the semiclassical limit, it is shown that, for energies where the classical system is completely chaotic, the level spacing statistics are Wigner, and where it is completely integrable, the level spacing statistics are Poisson. A model is described for the spacing distributions where the levels can be either Wigner or Poisson, which is useful for showing the transition from one to the other, and ad equately describes the statistics. Theoretically, the asymptotic level density for rotating billiards is calculated, and this is compared with the numerical results with good agreement, after modification of the method to include all levels.
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Tesař, Lukáš. "Nelineární dynamické systémy a chaos." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2018. http://www.nusl.cz/ntk/nusl-392844.
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The diploma thesis deals with nonlinear dynamical systems with emphasis on typical phenomena like bifurcation or chaotic behavior. The basic theoretical knowledge is applied to analysis of selected (chaotic) models, namely, Lorenz, Rössler and Chen system. The practical part of the work is then focused on a numerical simulation to confirm the correctness of the theoretical results. In particular, an algorithm for calculating the largest Lyapunov exponent is created (under the MATLAB environment). It represents the main tool for indicating chaos in a system.
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Kateregga, George William. "Bifurcations in a chaotic dynamical system." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2019. http://www.nusl.cz/ntk/nusl-401527.
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Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.
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Bick, Christian [Verfasser], Marc [Akademischer Betreuer] Timme, and Laurent [Akademischer Betreuer] Bartholdi. "Chaos and Chaos Control in Network Dynamical Systems / Christian Bick. Gutachter: Marc Timme ; Laurent Bartholdi. Betreuer: Marc Timme." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2013. http://d-nb.info/1044172517/34.
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Lan, Yueheng. "Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view." Diss., Available online, Georgia Institute of Technology, 2004:, 2004. http://etd.gatech.edu/theses/available/etd-10282004-154606/unrestricted/lan%5Fyueheng%5F200412%5Fphd.pdf.
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Thesis (Ph. D.)--Physics, Georgia Institute of Technology, 2005.Jean Bellissard, Committee Member ; Turgay Uzer, Committee Member ; Roman Grigoriev, Committee Member ; Konstantin Mischaikow, Committee Member ; Predrag Cvitanovic, Committee Chair. Vita. Includes bibliographical references.
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Zhang, Jianxin. "Investigation of chaos and nonlinear dynamical behaviour in self-driven oscillators." Thesis, Queen Mary, University of London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368957.
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Walker, Ricki Paterson. "Dynamical analysis of self-pulsation and chaos in semiconductor laser models." Thesis, Bangor University, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.421674.
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Mora, Karin. "Non-smooth dynamical systems and applications." Thesis, University of Bath, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636521.
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The purpose of this work is to illuminate some of the non-smooth phenomena found in piecewise-smooth continuous and discrete dynamical systems, which do not occur in smooth systems. We will explain how such non-smooth phenomena arise in applications which experience impact, such as impact oscillators, and a type of rotating machine, called magnetic bearing systems. The study of their dynamics and sensitivity to parameter variation gives not just insights into the critical motion found in these applications, but also into the complexity and beauty in their own right. This work comprises two parts. The first part studies a general one-dimensional discontinuous power law map which can arise from impact oscillators with a repelling wall. Parameter variation and the influence of the exponent on the existence and stability of periodic orbits is presented. In the second part we analyse two coupled oscillators that model rotating machines colliding with a circular boundary under friction. The study of the dynamics of rigid bodies impacting with and without friction is approached in two ways. On the one hand existence and stability conditions for non-impacting and impacting invariant sets are derived using local and global methods. On the other hand the analysis of parameter variation reveals new non-smooth bifurcations. Extensive numerical studies confirm these results and reveal further phenomena not attainable otherwise.
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Banks, Jess M. "Chaos and Learning in Discrete-Time Neural Networks." Oberlin College Honors Theses / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1445945609.
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Vasalos, Ioannis. "Transmission control protocol modelling of dynamical complexity and chaos in UMTS networks." Thesis, University of Newcastle Upon Tyne, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.590041.
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Teletraffic studies in wired networks showed that during peak traffic time chaotic dynamics appear in the network's traffic flows that can induce severe Quality of Service (QOS) degradation and destabilize network grids. The nature of these effects is targeted on the Transmission Control Protocol (TCP) as the dominant transmission protocol for data and Internet applications. These effects suggest that similar behaviour will appear in mobile networks of which the latest evolution is the Universal Mobile Telecommunications System (UMTS). So far there is still a limited understanding of UMTS traffic dynamics and performance under increased traffic loads and congestion. Therefore the objective of this Thesis is twofold. To study and analyze the nonlinear chaotic behaviour of the TCP inside the UMTS network along with the impact of this behaviour on the QoS. To present a modelling framework to portray the TCP traffic flows and capture the chaotic dynamics developed in the network traffic under increased congestion levels. Initially using the fundamentals of chaos theory chaotic dynamics in the TCP traffic flow are theoretically analysed by studying the TCP packet sending mechanism in relation to the feedback that TCP receives from the UMTS network. The theoretical analysis is validated with extensive simulation modelling of TCP traffic flows in the UMTS network under a variety of congestion level scenarios and taking into account the randomness existing in the feedback of the TCP from packet drops and packet delays inside the network. Analysis of the results is performed using the concepts of Lyapunov Exponents and exponential divergence curves as one of the strictest tests for deterministic chaos. Results showed that according to the network traffic load the TCP can exhibit from periodic to highly complex behaviour with a combination of dominant chaotic dynamics caused by the TCP and randomness induced from the network. This highly complex behaviour has a major impact on the QoS of the UMTS and results in unfairness and underutilization of the UMTS radio resources. From the theoretical model of the TCP and based on the hybrid systems modelling approach, a hybrid modelling platform is developed for modelling of the UMTS network. The model consists of continuous and discrete dynamical subsystems used to model the nonlinear dynamics of the TCP packet flow, the fading of the wireless radio channel, the Selective-Repeat Automatic Repeat Request Link Layer protocol and the packet flow in the queues of the network. The modelling implementation is extensively validated by comparison to OPNET simulations.
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Ji, Songkun. "Optical effects on the dynamical properties of semiconductor laser devices and their applications." Thesis, Bangor University, 2019. https://research.bangor.ac.uk/portal/en/theses/optical-effects-on-the-dynamical-properties-of-semiconductor-laser-devices-and-their-applications(fbf11b84-5465-4f2a-b029-ba8846082cad).html.
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Nonlinear dynamical properties of semiconductor lasers have attracted considerable attention, and their rich behaviors enable many popular research topics. The research effort of this thesis has emphasized on two areas - one is photonic microwave generation based on period one dynamic of semiconductor lasers; the other is laser's chaotic dynamic. Microwave photonics has attracted considerable attention recently because of its practical applications in radio-over-fiber (RoF) communications links. A stable photonic microwave allows it to convey, in a cost-effective manner, wideband signals over optical fibers with low loss, large bandwidth and immunity of electromagnetic interference. Microwave photonics technologies consist of photonic microwave generation, processing, control and distribution. Many photonic microwave generation techniques have been proposed, which includes direct modulation, optical heterodyne technique, external modulation, mode-locked semiconductor lasers, optoelectronic oscillator (OEO) and period one (P1) dynamic of semiconductor lasers. Among these techniques, photonic microwave generation based on P1 oscillation dynamic has gained special attention due to its many advantages, such as: widely tunable oscillation frequency, and nearly single sideband (SSB) spectrum. The aim of this thesis in the photonic microwave generation area is to produce photonic microwaves based on P1 dynamic using low-cost vertical-cavity surface-emitting lasers (VCSELs). The technical contents in this area cover two parts. The first part is to generate broadly tunable photonic microwaves. Continuous tuning of the microwave frequency from 4GHz to up to an instrumentation limited 15GHz is experimental achieved through the adjustment of the injection power and the frequency detuning between the master laser and the VCSEL. Numerical simulations using a common spin flip model are also carried out, which agree qualitatively with the experimental results. The second part of the photonic microwave generation in this thesis is to explore effective approaches to not only reduce the linewidth but also improve the stability of the generated microwave. Due to spontaneous emission noise in the semiconductor laser, P1 dynamic inherently imposes phase noise, which increases the microwave linewidth of the generated microwave. This considerably affect the signal transmission performance of the modulated microwave signal in RoF applications. To address this challenge, single optical feedback and double optical feedback are applied in the experiments. The experimental results demonstrate that both single feedback and double feedback can reduce the linewidth of the generated microwave to about one tenth of linewidth without the optical feedback. However, single optical feedback may induce many side peaks due to external cavity frequency from the feedback cavity, the feedback phase needs to be carefully adjusted to suppress the side peaks. The side peaks can be suppressed by introducing the second optical feedback. The double optical feedback can also significantly enhance the stability of the generated microwave. The results of the numerical simulations are in good agreement with the experimental results. The other important dynamic of semiconductor lasers is chaos, which has attracted considerable research interest due to its many potential applications in secure communications, chaotic optical time-domain reflectors, chaotic lidars and physical random number generators. Optical feedback is the simplest method to generate chaos in semiconductor lasers, but a typical chaos generated by optical feedback has unwanted recurrence features termed time delay (TD) signature because of the optical round trip in the external cavity. The complexity, bandwidth and TD signature of chaos are the three main parameters for evaluating its applicability in abovementioned application scenarios. In order to find the correct operating parameters to achieve low TD signature and high complexity of chaos simultaneously, in this thesis, the influence of bias current and the feedback strength on the complexity and time-delay signature of chaotic signals in semiconductor lasers with optical feedback is investigated experimentally and theoretically. In the experiment, the effect of the data acquisition method on quantification of complexity is also examined. The experimental results show that the TD signature is approximately in an inverse relationship with the complexity of chaos when the semiconductor laser is subject to low or strong optical feedback. However, the inverse relationship disappears when the laser operates at higher bias currents with intermediate feedback strength. Numerical simulation based on Lang Kobayashi laser equations show qualitative agreements with the experimental results.
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Khůlová, Jitka. "Stabilita a chaos v nelineárních dynamických systémech." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2018. http://www.nusl.cz/ntk/nusl-392836.
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Diplomová práce pojednává o teorii chaotických dynamických systémů, speciálně se pak zabývá Rösslerovým systémem. Kromě standardních výpočtů spojených s bifurkační analýzou se práce zaměřuje na problém stabilizace, konkrétně na stabilizaci rovnovážných bodů. Ke stabilizaci je využita základní metoda zpětnovazebního řízení s časovým zpožděním. Významnou část práce tvoří zavedení a implementace obecné metody pro hledání vhodné volby parametrů vedoucí k úspěšné stabiliaci. Dalším diskutovaným tématem je možnost synchronizace dvou Rösslerových systémů pomocí různých synchronizačních schémat.
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Kalamangalam, G. P. "Nonlinear oscillations and chaos in chemical cardiorespiratory control." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.296830.
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We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work.
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Prescott, Simon L. "Nonlinear dynamical analysis of electrocardiogram data and the prospects for control of cardiac chaos." Thesis, Keele University, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302274.
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Xu, Mu. "Spatiotemporal Chaos in Large Systems Driven Far-From-Equilibrium: Connecting Theory with Experiment." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/79499.
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There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity.Ph. D.
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Schneider, Judith. "Dynamical structures and manifold detection in 2D and 3D chaotic flows." Phd thesis, [S.l. : s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=973637420.
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Gotthans, Tomas. "Advanced methods for analyzing non-linear dynamical systems." Thesis, Paris Est, 2014. http://www.theses.fr/2014PEST1020/document.
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L'augmentation des performances des futurs systèmes dynamiques nécessite la prise en compte des phénomènes physiques non linéaires. Cette thèse apporte un éclairage et des contributions sur deux sujets complémentaires liés aux phénomènes dynamiques non linéaires. Le mémoire de thèse est divisé en deux parties.La première partie porte sur les non-linéarités des amplificateurs de puissance dans le cadre d'applications destinées aux télécommunications ou à la diffusion audio-visuelle. Plusieurs méthodes de modélisation et de linéarisation des amplificateurs de puissance ont été conçues et discutées. Un banc de test a été développé afin d'évaluer les méthodes sur des amplificateurs réels. La robustesse de ces techniques à un mauvais alignement temporel des signaux ainsi que leur capacité à faire face à des artefacts spectraux ont été évaluées. Par ailleurs, nous avons effectué une étude théorique sur l'existence et la prise en compte de solutions multiples dans l'approche adaptative par apprentissage indirect. La deuxième partie traite des systèmes dynamiques non linéaires qui présentent des solutions chaotiques. Ces systèmes sont bien connus, mais les techniques d'identification de ces solutions manquent de fiabilité ou nécessitent une puissance de calcul importante. Dans cette thèse, plusieurs méthodes utilisant également le calcul parallèle sont présentées. Les systèmes à commande différentielle fractionnaire sont brièvement discutés. Il est aussi montré, qu'il existe des systèmes liés à des fonctions de transfert non linéaires avec quantification pour lesquels les méthodes d'analyse classiques échouentIn order to achieve better performance of modern communication devices, that have to be operated on its physical limits, the nonlinear phenomena need to be taken into the account. This thesis brings insight into two different subjects related with nonlinear dynamical phenomena. The thesis itself is divided into two parts : the first part is focused on the domain of nonlinear power amplifiers from the system point of view. Several methods for modelization and linearization of power amplifiers have been designed and discussed. A test-bench has been assembled in order to evaluate the proposed methods on real power amplifiers. Then the robustness to time misalignment in the system and the ability to deal with spectral artifacts in the system of presented methods have been evaluated. Also a theoretical study has been conducted on the existence and management of multiple solutions in the frame of adaptive indirect learning approach. The second part deals with nonlinear dynamical systems that are exhibiting chaotic solutions. Such systems are well known, but techniques for identifying reliable such solutions are either missing or are computational intense. In this thesis several methods using also parallel computing are presented. Systems with fractional differential order are briefly discussed. It is as well shown, that there exists systems related with quantified nonlinear transfer functions for which the standard analyzing methods fails
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Yamanaka, Shogo. "Nonintegrability of Dynamical Systems near Equilibria and Heteroclinic Orbits." Kyoto University, 2020. http://hdl.handle.net/2433/253418.
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Dinius, Joseph. "Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems." Diss., The University of Arizona, 2014. http://hdl.handle.net/10150/315858.
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The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.
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Washburn, Auriel. "Harmony from Chaos? Investigations in Aperiodic Visual-Motor and Interpersonal Coordination." University of Cincinnati / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1397733911.
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Shea-Blymyer, Colin Russel. "Distinguishing Dynamical Kinds: An Approach for Automating Scientific Discovery." Thesis, Virginia Tech, 2019. http://hdl.handle.net/10919/101659.
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The automation of scientific discovery has been an active research topic for many years.
The promise of a formalized approach to developing and testing scientific hypotheses has
attracted researchers from the sciences, machine learning, and philosophy alike. Leveraging
the concept of dynamical symmetries a new paradigm is proposed for the collection of
scientific knowledge, and algorithms are presented for the development of EUGENE – an
automated scientific discovery tool-set. These algorithms have direct applications in model
validation, time series analysis, and system identification. Further, the EUGENE tool-set
provides a novel metric of dynamical similarity that would allow a system to be clustered
into its dynamical regimes. This dynamical distance is sensitive to the presence of chaos,
effective order, and nonlinearity. I discuss the history and background of these algorithms,
provide examples of their behavior, and present their use for exploring system dynamics.Master of Science
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Polo, Fabrizio. "Equidistribution on Chaotic Dynamical Systems." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1306527005.
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Tufaile, Alberto. "Estudo da formação de bolhas em líquidos viscosos (uma abordagem usando a teoria do caos)." Universidade de São Paulo, 2000. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-02122013-190512/.
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Construímos um aparato experimental para estudar a dinâmica da formação de bolhas de ar em um bico submerso em uma solução de água/glicerina dentro de um tubo cilíndrico. O tempo entre bolhas sucessivas foi medido com um sistema laser/fotodiodo. Os resultados experimentais foram interpretados usando a Teoria do Caos. Foram observados bifurcações, comportamento caótico e saltos no regime periódico, em função da diminuição da vazão do ar soprado no bico. Além das transições dos regimes do borbulhamento, nós também observamos efeitos na dinâmica do borbulhamento quando aplicamos uma onda sonora sintonizada na frequência fundamental da coluna de ar acima do líquido onde as bolhas eram formadas. Em função da amplitude da onda sonora, nós obtivemos ciclo limite, bifurcação flip, comportamento caótico e sincronização do borbulhamento com a frequência da onda sonora. Utilizando caracterizações métrica e topológica em alguns atratores, pudemos relacioná-los com uma dinâmica tipo-Hénon cujo comportamento é um caso particular da dinâmica do mapa do círculo bidimensional. Observamos características presentes na dinâmica do mapa do círculo na formação das bolhas variando a amplitude da onda sonora, tais como transição para o Caos via quase-periodicidade, cascata de duplicações de período e Caos.We have constructed an experimental apparatus to study the dynamics of the formation of air bubbles in a nozzle submerged in a water/glycerin solution inside a cy1indrical tube. The time delay between successive bubbles was measured with a laser/photodiode system. The results were interpreted by means of Chaos Theory, and it was observed bífurcations, chaotic behavior, and sudden changes in a periodic regime as a function of decreasing air flow rate issued through the nozzle. Besides bubbling regime transitions, we also observed dynamical effects by applying a sound wave tuned to the fundamental frequency of the air column above the liquid of the bubble formation, As a function of the sound wave amplitude. we obtained limit cycle, flip bifurcation, chaotic behavior, and synchronization of the bubbling with the sound wave frequency. Applying metrical as well as topological characterization to some chaotic attractors, we could establish relation with a Hénon-like dynamics. The Hénon-like behavior is a particular case of the dissipative two-dimenslonal circle-rnap dynamics, and by varying the amplitude of a sound wave, we have observed featutes present in the circle map dynamics, such as transition from quasiperiodic to chaotic behavior, period doubling cascade, and Chaos.
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Löck, Steffen. "Dynamical Tunneling and its Application to Spectral Statistics." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-159816.
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Tunneling is a central result of quantum mechanics. It allows quantum particles to enter regions which are inaccessible by classical dynamics. Consequences of the tunneling process are most relevant. For example it causes the alpha-decay of radioactive nuclei and it is argued that proton tunneling is decisive for the emergence of DNA mutations. The theoretical prediction of corresponding tunneling rates is explained in standard textbooks on quantum mechanics for regular systems. Typical physical systems such as atoms or molecules, however, also show chaotic motion. Here the calculation of tunneling rates is more demanding. In this text a selection of articles on the prediction of tunneling rates in systems which allow for regular and chaotic motion is summarized. The presented approach is then used to explain consequences of tunneling on the quantum spectrum, such as the universal power-law behavior of small energy spacings and the flooding of regular states.
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Fasoli, Ambrogio. "Study of wave-particle interaction from the linear regime to dynamical chaos in a magnetized plasma /." [S.l.] : [s.n.], 1993. http://library.epfl.ch/theses/?nr=1162.
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Tang, Man. "A study of the nonlinear dynamics nature of ECG signals using Chaos theory." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B34624843.
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Tang, Man, and 鄧敏. "A study of the nonlinear dynamics nature of ECG signals using Chaos theory." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2005. http://hub.hku.hk/bib/B34624843.
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Isliker, Heinz. "Dynamical properties of bursts and flares : an inquiry on deterministic chaos in the solar and stellar coronae /." [S.l.] : [s.n.], 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10495.
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Schick, Michael [Verfasser], and V. [Akademischer Betreuer] Heuveline. "Uncertainty Quantification for Stochastic Dynamical Systems : Spectral Methods using Generalized Polynomial Chaos / Michael Schick. Betreuer: V. Heuveline." Karlsruhe : KIT-Bibliothek, 2012. http://d-nb.info/101936193X/34.
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Brandon, Quentin. "Numerical method of bifurcation analysis for piecewise-smooth nonlinear dynamical systems." Toulouse, INSA, 2009. http://eprint.insa-toulouse.fr/archive/00000312/.
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Dans le domaine de l’analyse des systèmes dynamiques, les modèles lisses par morceaux ont gagné en popularité du fait de leur grande flexibilité et précision pour la représentation de certains systèmes dynamiques hybrides dans des applications telles que l’électronique ou la mécanique. Les systèmes dynamiques hybrides possèdent deux ensembles de variables, l’un évoluant dans un espace continu, l’autre dans un espace discret. La plupart des méthodes d’analyse nécessitent que l’orbite reste lisse pour être applicable, de telle sorte que certaines manipulations d’adaptation aux systèmes hybrides deviennent inévitables lors de leur analyse. Sur la base d’un modèle lisse par morceaux, où l’orbite du système est découpée en morceaux localement lisses, et une méthode d’analyse des bifurcations hybride, utilisant une application de Poincaré dont les sections sont régies par les conditions de commutation du système, nous étudions le processus d’analyse en détails. Nous analysons ensuite plusieurs extensions de l’oscillateur d’Aplazur, dont la version originale est un oscillateur bidimensionnel non-lisse à commutation. Ce dernier, en tant que système dynamique non linéaire à commutation, est un excellent candidat pour démontrer l’efficacité de cette approche. De plus, chaque extension présente un nouveau scénario, permettant d’introduire les démarches appropriées et d’illustrer la flexibilité du modèle. Finalement, afin d’exposer l’implémentation de notre programme, nous présentons quelques unes des méthodes numériques les plus pertinentes. Il est intéressant de signaler que nous avons choisi de mettre l’accent sur les systèmes dynamiques autonomes car le traitement des systèmes non-autonomes nécessitent seulement une simplification (pas de variation du temps). Cette étude présente une méthode généraliste et structurée pour l’analyse des bifurcations des systèmes dynamiques non-linéaires hybrides, illustrée par des résultats pratiques. Parmi ces derniers, nous exposons quelques propriétés locales et globales de l’oscillateur d’Alpazur, dont la présence d’une cascade de points cuspidaux dans le diagramme de bifurcation. Notre travail a abouti à la réalisation d’un outil d’analyse informatique, programmé en C++, utilisant les méthodes numériques que nous avons sélectionnées à cet effet, telles que l’approximation numérique de la dérivée seconde des éléments de la matrice JacobienneIn the field of dynamical system analysis, piecewise-smooth models have grown in popularity due to there greater flexibility and accuracy in representing some hybrid systems in applications such as electronics or mechanics. Hybrid dynamical systems have two sets of variables, one which evolve in a continuous space, and the other in a discrete one. Most analytical methods require the orbit to be smooth during objective intervals, so that some special treatments are inevitable to study the existence and stability of solutions in hybrid dynamical systems. Based on a piecewise-smooth model, where the orbit of the system is broken down into locally smooth pieces, and a hybrid bifurcation analysis method, using a Poincare map with sections ruled by the switching conditions of the system, we review the analysis process in details. Then we apply it to various extensions of the Alpazur oscillator, originally a nonsmooth 2-dimension switching oscillator. The original Alpazur oscillator, as a simple nonlinear switching system, was a perfect candidate to prove the efficiency of the approach. Each of its extensions shows a new scenario and how it can be handled, in order to illustrate the generality of the model. Finally, and in order to show more of the implementation we used for our own computer-based analysis tool, some of the most relevant numerical methods we used are introduced. It is noteworthy that the emphasis has been put on autonomous systems because the treatment of non-autonomous ones only requires a simplification (no time variation). This study brings a strong and general framework for the bifurcation analysis of nonlinear hybrid dynamical systems, illustrated by some results. Among them, some interesting local and global properties of the Alpazur Oscillator are revealed, such as the presence of a cascade of cusps in the bifurcation diagram. Our work resulted in the implementation of an analysis tool, implemented in C++, using the numerical methods that we chose for this particular purpose, such as the numerical approximation of the second derivative elements in the Jacobian matrix
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Ahlers, Volker. "Scaling and synchronization in deterministic and stochastic nonlinear dynamical systems." Phd thesis, [S.l.] : [s.n.], 2001. http://pub.ub.uni-potsdam.de/2002/0001/ahlers.pdf.
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Teles, Renato de Sá 1972. "Bilhares : aspectos clássicos e quânticos." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307108.
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Orientador: Alberto Vazquez SaaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-20T15:42:31Z (GMT). No. of bitstreams: 1 Teles_RenatodeSa_D.pdf: 3942109 bytes, checksum: 7e41b541aa6eb7a186a4956d751a32c5 (MD5) Previous issue date: 2012
Resumo: Fizemos um estudo sistemático dos aspectos clássicos e quânticos dos sistemas dinâmicos conhecidos como "bilhares". Introduzimos uma nova classe de bilhares classicamente caóticos cuja dinâmica quântica pode ser convenientemente descrita utilizando-se uma aproximação do tipo Galerkin, o que nos permitiu obter com boa precisão um grande numero de autovalores e autofunções e estudar algumas propriedades estatísticas do espectro de energia para esta nova classe de bilhares. Do ponto de vista da implementação numérica, estudamos também os efeitos de tamanho finito da matriz associada ao truncamento dos modos de Galerkin
Abstract: We consider classical and quantum aspects of the dynamical systems dubbed as "billiards". We introduce a new class of classically chaotic billiards for which the quantum dynamics can be conveniently described by a Galerkin type approximation, allowing us to obtain with good accuracy a large number of eigenvalues and eigenfunctions and to study some statistical properties of the energy spectrum of this new class of billiards. From the numerical implementation point of view, we consider also the finite size effects on the matrix corresponding to the truncation of the Galerkin modes
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
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Nascimento, Roberto Venegeroles. "Teoria cinética de mapas hamiltonianos." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-29022008-115433/.
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Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricasThis work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations
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Funabashi, Masatoshi. "Dynamical System and Information Geometry : A Complementary Approach to Complex Systems." Palaiseau, Ecole polytechnique, 2010. http://pastel.archives-ouvertes.fr/docs/00/55/68/73/PDF/thesis.pdf.
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En français : Un des défis majeurs de la science de complexité se situe à l'investigation de l'émergence, où les interactions entre les composants microscopiques d'un système produisent la propriété globale, et réciproquement, la dynamique globale influence le bas niveau. Cette thèse a comme ambition de 1) élucider le mécanisme sous-jacent des systèmes complexes par la modélisation concrète des systèmes réels, et aussi 2) comparer entre les différents modèles proposés pour détecter la condition universelle de l'émergence. Pour cela, nous développons la nouvelle méthodologie basé sur l'interaction entre la théorie de système dynamique et la géométrie informationnelle, afin d'avoir la dialectique entre la modélisation constructive/déterministe et l'analyse des interactions sous la formalisation stochastique. La thèse se compose de 7 Parties, parmi lesquelles la Partie 2 à 6 correspondent au premier objectif, et la Partie 7 au seconde. Dans la Partie 1, nous allons réviser l'histoire de la science de la complexité et proposer la stratégie dialectique entre les méthodologies constructive et interactions-analytique, basé sur la théorie de système dynamique et la géométrie informationnelle. En Partie 2, nous traitons un modèle de réseau neuronal avec le comportement chaotique nommé ``l'itinérance chaotique" comme un candidat de la dynamique du cortex cérébral, et analysons l'effet de l'apprentissage autonome sans superviseur comme une source de créativité qui est la propriété émergente du système neuronal. La théorie de la mesure intérieure est étendue afin de interpréter l'émergence des nouveaux attracteurs par ``le chaos comme le catalyseur d'apprentissage. " En Partie 3, nous avons appliqué la dynamique du réseau neuronal chaotique aux robots qui manifestent la dynamique de recherche collective de manière émergente, au défi de la détection optimale des informations sporadiques. L'efficacité de la recherche collective est évaluée avec un simulateur virtuel. En Partie 4, nous développons les nouvelles mesures de la complexité du point de vue de la géométrie informationnelle, et analysons les données des réseaux sociaux. Les mesures de la complexité jouera un rôle principal dans la Partie 7. En Partie 5, nous appliquons la stratégie dialectique entre le système dynamique et la géométrie informationnelle vers la compréhension de la morphogenèse lors de l'embryogenèse chez le poisson zèbre. Quelques propositions théoriques sont établies et testées avec les données tentatives dérivées des projets européens Embryomics et BioEMERGENCES. En Partie 6, nous analysons les systèmes complexes liés au linguistique. Nous avons découvert les nouveaux invariants et la composition géométrique entre les voyelles japonaises, qui sont les propriétés émergentes au niveau du système. Nous développons aussi la modélisation écologique de l'environnement multilingue dans un contexte de la dialectique entre la théorie linguistique et la modélisation mathématique. En Partie 7, nous révisons les résultats obtenus dans les Parties précédentes sous une perspective comparative, en vue de détecter la structure universelle de l'émergence comme l'organisation des interactions qui ne dépende pas explicitement sur la propriété des composants. Surtout la comparaison entre les Parties 2 et 4, ainsi 5 et 6, nous indique la typologie et la stratégie de détection de la dynamique de l'émergence comme la relation et le contraint entre les foncteurs et méta-foncteurs. D'autre possibilité d'application de la stratégie établie est mise en discussionEn anglais : Recently emerging complex systems sciences tackle the systems where complex in- teractions between components lead to the manifestation of emergent property linking different levels of organization. This thesis aims to reveal the mechanism of emergent property in complex systems, both in concrete modeling as well as comparative analysis between different systems. We tackle various sub jects in complex systems science with newly proposed unified theoretical framework, based on the dialectic between dynam- ical system theory and information geometry. The thesis has therefore two levels of ob jectives: 1) Modeling and understanding of concrete complex systems with the use of constructive and interaction-analytical methodologies, and 2) comparison between different complex systems to characterize universal structure of emergence. The thesis consists of 7 Parts, in which Part 2 to 6 correspond to the first ob jective, and the Part 7 to the second one: In Part 1, we review the historical context of complex systems science and propose a dialectical strategy between the constructive and interaction-analytical methodology, based on the dynamical system theory and information geometry, respectively. In Part 2, we treat a candidate model of brain cortex dynamics known as “chaotic itinerancy”, and incorporate the effect of autonomous learning seeking for the creativity of intelligence as emergent property of neural system. The interpretation of emergence in terms of the internal measurement theory is extended to derive the concept of “chaotic itinerancy as catalyst of learning”. In Part 3, the dynamics of chaotic neural network is applied to emergent collective behavior of robots, so that to realize optimal intermittent search of sporadic informa- tion. The effectiveness of the collective infotaxis is analyzed on a simulator basis. In Part 4, we define novel complexity measures from information geometrical point of view and apply to the analysis of social network data. The established complexity measures play a key role in comparative analysis between different systems in Part 7. In Part 5, we apply the dialectical strategy between dynamical system and infor- mation geometry toward the understanding of morphogenesis during zebrafish embryo- genesis. Theoretical propositions are tested with tentative experimental data from two european pro jects, Embryomics and BioEMERGENCES. In Part 6, complex systems related to linguistics are investigated. We discovered novel invariants and geometrical relation between japanese vowels, as a system-level emergent property. Ecological modeling approach to multilingual environment is also proposed along the dialectical strategy between linguistic theory and mathematical modeling. In Part 7, we review the obtained results in previous Parts with comparative per- spective, seeking for a characterization of universal structure of emergence in terms of the organization of interactions that does not explicitly depend on the property of components. Comparison between Part 2 and 4, as well as 5 and 6, derived candi- date qualitative dynamics of emergence and its detection strategy as the dynamics and constraint between functors and meta-functors. Further possibility of the proposed strategy is discussed
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Le, Sceller Lois. "Reconstruction globale de champ de vecteurs et applications." Rouen, 1997. http://www.theses.fr/1997ROUES012.
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La reconstruction globale de champ de vecteurs est une technique de modélisation phénoménologique de systèmes dynamiques déterministes à faible nombre de degrés de liberté. La méthode de reconstruction globale présentée dans ce manuscrit consiste à construire un système d'équations différentielles ordinaires à partir de la connaissance de l'évolution d'une observable au cours du temps. L'intégration du modèle ainsi obtenu fournit alors une reproduction du comportement dynamique de la variable observée. Cette technique et les améliorations qui lui ont été apportées au cours de ce travail y sont décrites, ainsi que les outils numériques mis au point afin de la rendre applicable à la plus vaste gamme de signaux possibles issus de systèmes expérimentaux. Sont ensuite proposés des exemples d'applications, à des signaux issus, i) de systèmes théoriques : Lorenz, Rossler, Burke & Shaw, système hyperchaotique de Rossler et équations de Duffing, ii) de systèmes expérimentaux : deux réactions de Belousov-Zhabotinskii, deux réactions d'électrodissolution (du cuivre et du fer), des instabilités de fil chaud et de lentille thermique, et un écoulement turbulent dans une cuve de mélange. Enfin, une extension de la méthode permettant d'inclure au modèle une dépendance explicite à des paramètres de contrôle du système observé est proposée. Une fois la faisabilité de cette dernière démontrée sur le système de Rossler, l'application de cette technique de modélisation est réalisée à partir des évolutions temporelles du courant mesurées sur une expérience d'électrodissolution du cuivre pour différentes valeurs du potentiel aux électrodes.
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Passey, Jr David Joseph. "Growing Complex Networks for Better Learning of Chaotic Dynamical Systems." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8146.
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This thesis advances the theory of network specialization by characterizing the effect of network specialization on the eigenvectors of a network. We prove and provide explicit formulas for the eigenvectors of specialized graphs based on the eigenvectors of their parent graphs. The second portion of this thesis applies network specialization to learning problems. Our work focuses on training reservoir computers to mimic the Lorentz equations. We experiment with random graph, preferential attachment and small world topologies and demonstrate that the random removal of directed edges increases predictive capability of a reservoir topology. We then create a new network model by growing networks via targeted application of the specialization model. This is accomplished iteratively by selecting top preforming nodes within the reservoir computer and specializing them. Our generated topology out-preforms all other topologies on average.
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Cha, Gun-Ho. "Reappraisal of market efficiency tests arising from nonlinear dependence, fractals, and dynamical systems theory." Doctoral thesis, Stockholm : Economic Research Institute, Stockholm School of Economics [Ekonomiska forskningsinstitutet vid Handelshögsk.] (EFI), 1993. http://www.hhs.se/efi/summary/365.htm.
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Marin, Boris. "Determinismo e estocasticidade em modelos de neurônios biológicos." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-23092014-154612/.
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Investigou-se a gênese de atividade irregular em neurônios de centros geradores de padrões através de modelos eletrofisiologicamente realistas. Para tanto, foram adotadas abordagens paralelas. Primeiramente, desenvolveram-se técnicas para determinar quais os mecanismos biofísicos subjacentes aos processos de codificação de informação nestas células. Também foi proposta uma nova metodologia híbrida (baseada em continuação numérica e em varreduras força bruta) para análise de bancos de dados de modelos neuronais, permitindo estendê-los e revelar instâncias de multiestabilidade entre regimes oscilatórios e quiescentes. Além disto, a fim de determinar a origem de comportamento complexo em modelos neuronais simplificados, empregaram-se métodos geométricos da teoria de sistemas dinâmicos. A partir da análise de mapas unidimensionais perturbados por ruído, foram discutidos possíveis cenários para o surgimento de caos em sistemas dinâmicos aleatórios. Finalmente mostrou-se que, levando em conta o ruído, uma classe de modelos de condutâncias reproduz padrões de disparo observados in vivo. Estas pertubações revelam a riqueza da dinâmica transiente, levando o sistema a visitar um arcabouço determinista complexo preexistente -- sem recorrer a ajustes finos de parâmetros ou a construções ad hoc para induzir comportamento caótico.We investigated the origin of irregularities in the dynamics of central pattern generator neurons, through analyzing electrophysiologically realistic models. A number of parallel approaches were adopted for that purpose. Initially, we studied information coding processes in these cells and proposed a technique to determine the underlying biophysical mechanisms. We also developed a novel hybrid method (based on numerical continuation and brute force sweeps) to analyze neuronal model databases, extending them and unveiling instances of multistability between oscillatory and resting regimes. Furthermore, in order to determine the origin of irregular dynamics in simplified neuronal models, we employed geometrical methods from the theory of dynamical systems. The analysis of stochastically perturbed maps allowed us to discuss possible scenarios for the generation of chaotic behaviour in random dynamical systems. Finally we showed that, by taking noise into account, a class of conductance based models gives rise to firing patterns akin to the ones observed \\emph{in vivo}. These perturbations unveil the richness of the transient dynamics, inducing the system to populate a preexistent complex deterministic scaffolding -- without resorting to parameter fine-tuning or ad hoc constructions to induce chaotic activity.
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Blackbeard, Nicholas. "A journey through the dynamical world of coupled laser oscillators." Thesis, University of Exeter, 2012. http://hdl.handle.net/10036/3593.
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The focus of this thesis is the dynamical behaviour of linear arrays of laser oscillators with nearest-neighbour coupling. In particular, we study how laser dynamics are influenced by laser-coupling strength, $\kappa$, the natural frequencies of the uncoupled lasers, $\tilde{\Omega}_j$, and the coupling between the magnitude and phase of each lasers electric field, $\alpha$. Equivariant bifurcation analysis, combined with Lyapunov exponent calculations, is used to study different aspects of the laser dynamics. Firstly, codimension-one and -two bifurcations of relative equilibria determine the laser coupling conditions required to achieve stable phase locking. Furthermore, we find that global bifurcations and their associated infinite cascades of local bifurcations are responsible for interesting locking-unlocking transitions. Secondly, for large $\alpha$, vast regions of the parameter space are found to support chaotic dynamics. We explain this phenomenon through simulations of $\alpha$-induced stretching-and-folding of the phase space that is responsible for the creation of horseshoes. A comparison between the results of a simple {\it coupled-laser model} and a more accurate {\it composite-cavity mode model} reveals a good agreement, which further supports the use of the simpler model to study coupling-induced instabilities in laser arrays. Finally, synchronisation properties of the laser array are studied. Laser coupling conditions are derived that guarantee the existence of synchronised solutions where all the lasers emit light with the same frequency and intensity. Analytical stability conditions are obtained for two special cases of such laser synchronisation: (i) where all the lasers oscillate in-phase with each other and (ii) where each laser oscillates in anti-phase with its direct neighbours. Transitions from complete synchronisation (where all the lasers synchronise) to optical turbulence (where no lasers synchronise and each laser is chaotic in time) are studied and explained through symmetry breaking bifurcations. Lastly, the effect of increasing the number of lasers in the array is discussed in relation to persistent optical turbulence.
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Poignard, Camille. "Sur la synchronisation et la désynchronisation des systèmes dynamiques. Applications." Thesis, Nice, 2013. http://www.theses.fr/2013NICE4041/document.
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Cette thèse traite de la synchronisation et de la désynchronisation des systèmes dynamiques. Dans une première partie nous abordons, sous l’angle de la biologie systémique, le problème de la désynchronisation qui consiste à induire un comportement chaotique dans un système ayant une dynamique stable. Nous étudions ce problème sur un réseau génétique appelé V-système, inventé afin de coupler le plus simplement possible une bifurcation de Hopf et une hystérèse. Après avoir démontré qu’un champ de vecteurs de R^n présentant un tel couplage peut, sous certaines conditions, avoir un comportement chaotique, nous donnons un ensemble de paramètres pour lequel le V-système associé satisfait ces conditions et vérifions numériquement que le mécanisme responsable du chaos prend place dans ce système. Dans une deuxième partie, nous nous intéressons à la synchronisation de systèmes organisés hiérarchiquement. Nous commençons par définir une structure hiérarchique pour un ensemble de 2^n systèmes par une matrice représentant les étapes d’un processus de regroupement deux par deux. Cela nous amène naturellement au cas d’un ensemble de Cantor de systèmes, pour lequel nous obtenons un résultat de synchronisation globale généralisant le cas fini. Enfin nous traitons de la situation où certains défauts apparaissent dans la hiérarchie, i.e que certains liens entre les systèmes sont brisés. Nous montrons que l’on peut accepter un nombre infini de liens brisés, tout en gardant une synchronisation locale, à condition que ces liens soient uniquement présents aux N premiers étages de la hiérarchie (pour un N fixé) et qu’ils soient suffisamment espacés dans ces étagesThis thesis deals with the synchronization and desynchronization of dynamical systems. In a first part we tackle (under a biological viewpoint) the desynchronization problem, which consists in the induce- ment of a chaotic behavior in a stable dynamical system. We study this problem on a gene regulatory network called V-system, invented in order to couple in a very simple way, a Hopf bifurcation and a hysteresis-type dynamics. After having proved that a vector field on Rn admitting such a coupling may, under some condi- tions, show a chaotic dynamics, we give a set of parameters for which the associated V-system satisfies these conditions and verify numerically that the mechanism responsible of the chaotic motion occurs in this system. In a second part, we take interest in the synchronization of hierarchically organized dynamical systems. We first define a hierarchical structure for a set of 2^n systems by a matrix representing the steps of a matching process in groups of size two. This leads us naturally to the case of a Cantor set of systems, for which we obtain a global synchronization result generalizing the finite case. Finally, we deal with the situation where some defects appear in the hierarchy, that is to say when some links between certain systems are broken. We prove we can afford an infinite number of such broken links while keeping a local synchronization, providing they are only present at the first N stages of the hierarchy (for a fixed integer N) and they are enough spaced out in these stages
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Kurskoy, Yu S., O. S. Hnatenko, Yu P. Machekhin, and M. V. Neofitnyy. "Topological Model of Laser Emission Parameters Research." Thesis, CAOL, 2019. http://openarchive.nure.ua/handle/document/15100.
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The research paper presents a model for studying both the parameters and dynamics of laser light as a nonlinear dynamic system. The model provides for the measurement of the values of physical quantities by non-linear metrology methods and the analysis of the research findings with topological tools. The model is based on the assumption of interval values of the measured values and the possibility of changing the stationary dynamics into the random one. The model contains an experiment scheme and a procedure for evaluating measurement results. The peculiarity of the model lies in its systemic approach and suitability for measuring and researching stationary and chaotic modes. The model provides for the measurement of the emission parameter values intervals in various modes, of their stability values and time series prediction. Classification of the system dynamics is performed using the fractal dimension. The model can be used both to ensure the stability of the laser light parameters, and to obtain and control random emission.
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Chousionis, Vasileios. "Thermodynamical Formalism." Thesis, University of North Texas, 2004. https://digital.library.unt.edu/ark:/67531/metadc4631/.
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Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
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Ruiz, Mora Africa. "Bounded Surfatron Acceleration in the Presence of Random Fluctuations." Master's thesis, Temple University Libraries, 2015. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/345317.
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Mechanical EngineeringM.S.M.E.
The mechanisms of acceleration and transport of collisionless plasma in the presence of electromagnetic turbulence (EMT) still remains not fully understood. The particle-EMT interaction can be modelled as the interaction of the particle with a particular wave in the presence of random noise. It has been shown that in such a model the acceleration of the charged particles can be almost free. This effect is known as resonance, which can be explained by the so-called “surfatron” mechanism. We have conducted several numerical simulations for the models with and without the presence of EMT. The turbulence has been modeled as small random fluctuations on the background magnetic field. Particles dynamics consist of two regimes of motion: (i) almost free (Larmor) rotation and (ii) captured (resonance) propagation, which are given by two different sets of invariants. We have determined the necessary conditions for capture and release from resonance for the model without fluctuations, as well as the intrinsic structure of the initial conditions domain for particles in order to be captured. We observed a difference in the orders of magnitude of the dispersion of adiabatic invariant due to the effects of the added fluctuations at the resonance. These results are important to describe the mixing of the different energy levels in the presence of EMT. To understand the impact of the EMT on the system dynamics, we have performed statistical analysis of the effects that different characteristics of the random fluctuations have on the system. The particles' energy gain can be viewed as a random walk over the energy levels, which can be described in terms of the diffusion partial differential equation for the probability distribution function. This problem can be reverse-engineered to understand the nature and structure of the EMT, knowing beforehand the energy distribution of a set of particles.
Temple University--Theses