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Dissertations / Theses on the topic 'Dynamical Systems'
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1
Behrisch, Mike, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, and Stefan Siegmund. "Dynamical Systems in Categories." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-129909.
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In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.
2
Zaks, Michael. "Fractal Fourier spectra in dynamical systems." Thesis, [S.l.] : [s.n.], 2001. http://pub.ub.uni-potsdam.de/2002/0019/zaks.ps.
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Haydn, Nicolai Theodorus Antonius. "On dynamical systems." Thesis, University of Warwick, 1986. http://wrap.warwick.ac.uk/55813/.
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Part A. We prove existence of smooth invariant circles for area preserving twist maps close enough to integrable using renormalisation. The smoothness depends upon that of the map and the Liouville exponent of the rotation number. Part B. Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable function there on are known to be Gibbs states. The converse in discussed in this paper, where the problem is reduced to shift spaces and there solved by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known. Part C. On subshifts which are derived from Markov partitions exists an equivalence relation which idendifies points that lie on the boundary set of the partition. In this paper we restrict to symbolic dynamics. We express the quotient space in terms of a non-transitive subshift of finite type, give a necessary and sufficient condition for the existence of a local product structure and evaluate the Zeta function of the quotient space. Finally we give an example where the quotient space is again a subshift of finite type.
4
Miles, Richard Craig. "Arithmetic dynamical systems." Thesis, University of East Anglia, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323222.
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5
Che, Dzul-Kifli Syahida. "Chaotic dynamical systems." Thesis, University of Birmingham, 2012. http://etheses.bham.ac.uk//id/eprint/3410/.
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In this work, we look at the dynamics of four different spaces, the interval, the unit circle, subshifts of finite type and compact countable sets. We put our emphasis on chaotic dynamical system and exhibit sufficient conditions for the system on the interval, the unit circle and subshifts of finite type to be chaotic in three different types of chaos. On the interval, we reveal two weak conditions’s role as a fast track to chaotic behavior. We also explain how a strong dense periodicity property influences chaotic behavior of dynamics on the interval, the unit circle and subshifts of finite type. Finally we show how dynamics property of compact countable sets effecting the structure of the sets.
6
Hillman, Chris. "Sturmian dynamical systems /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/5806.
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Umenberger, Jack. "Convex Identifcation of Stable Dynamical Systems." Thesis, The University of Sydney, 2017. http://hdl.handle.net/2123/17321.
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This thesis concerns the scalable application of convex optimization to data-driven modeling of dynamical systems, termed system identi cation in the control community. Two problems commonly arising in system identi cation are model instability (e.g. unreliability of long-term, open-loop predictions), and nonconvexity of quality-of- t criteria, such as simulation error (a.k.a. output error). To address these problems, this thesis presents convex parametrizations of stable dynamical systems, convex quality-of- t criteria, and e cient algorithms to optimize the latter over the former. In particular, this thesis makes extensive use of Lagrangian relaxation, a technique for generating convex approximations to nonconvex optimization problems. Recently, Lagrangian relaxation has been used to approximate simulation error and guarantee nonlinear model stability via semide nite programming (SDP), however, the resulting SDPs have large dimension, limiting their practical utility. The rst contribution of this thesis is a custom interior point algorithm that exploits structure in the problem to signi cantly reduce computational complexity. The new algorithm enables empirical comparisons to established methods including Nonlinear ARX, in which superior generalization to new data is demonstrated. Equipped with this algorithmic machinery, the second contribution of this thesis is the incorporation of model stability constraints into the maximum likelihood framework. Speci - cally, Lagrangian relaxation is combined with the expectation maximization (EM) algorithm to derive tight bounds on the likelihood function, that can be optimized over a convex parametrization of all stable linear dynamical systems. Two di erent formulations are presented, one of which gives higher delity bounds when disturbances (a.k.a. process noise) dominate measurement noise, and vice versa. Finally, identi cation of positive systems is considered. Such systems enjoy substantially simpler stability and performance analysis compared to the general linear time-invariant iv Abstract (LTI) case, and appear frequently in applications where physical constraints imply nonnegativity of the quantities of interest. Lagrangian relaxation is used to derive new convex parametrizations of stable positive systems and quality-of- t criteria, and substantial improvements in accuracy of the identi ed models, compared to existing approaches based on weighted equation error, are demonstrated. Furthermore, the convex parametrizations of stable systems based on linear Lyapunov functions are shown to be amenable to distributed optimization, which is useful for identi cation of large-scale networked dynamical systems.
8
Freeman, Isaac. "A modular system for constructing dynamical systems." Thesis, University of Canterbury. Mathematics, 1998. http://hdl.handle.net/10092/8888.
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This thesis discusses a method based on the dual principle of Rössler, and developed by Deng, for systematically constructing robust dynamical systems from lower dimensional subsystems. Systems built using this method may be modified easily, and are suitable for mathematical modelling. Extensions are made to this scheme, which allow one to describe a wider range of dynamical behaviour. These extensions allow the creation of systems that reproduce qualitative features of the Lorenz Attractor (including bifurcation properties) and of Chua's circuit, but which are easily extensible.
9
Ozaki, Junichi. "Dynamical quantum effects in cluster dynamics of Fermi systems." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199083.
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CAPPELLINI, VALERIO. "QUANTUM DYNAMICAL ENTROPIES AND COMPLEXITY IN DYNAMICAL SYSTEMS." Doctoral thesis, Università degli studi di Trieste, 2004. http://thesis2.sba.units.it/store/handle/item/12545.
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2002/2003<br>We analyze the behavior of two quantum dynamical entropies in
connection with the classical limit. Using strongly chaotic
classical dynamical systems as models (Arnold Cat Maps and Sawtooth
Maps), we also propose a discretization procedure that resembles
quantization; even in this case, studies of quantum dynamical entropy
production are carried out and the connection with the continuous
limit is explored. In both case (quantization and discretization) the
entropy production converge to the Kolmogorov-Sinai invariant on
time-scales that are logarithmic in the quantization (discretization)
parameter.<br>XVI Ciclo<br>1969<br>Versione digitalizzata della tesi di dottorato cartacea.
11
McKee, Andrew. "Multipliers of dynamical systems." Thesis, Queen's University Belfast, 2017. https://pure.qub.ac.uk/portal/en/theses/multipliers-of-dynamical-systems(65b93a06-6e7b-420b-ae75-c28d373f8bdf).html.
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Herz–Schur multipliers of a locally compact group have a well developed theory coming from a large literature; they have proved very useful in the study of the reduced C∗-algebra of a locally compact group. There is also a rich connection to Schur multipliers,which have been studied since the early twentieth century, and have a large number of applications. We develop a theory of Herz–Schur multipliers of a C∗-dynamical system, extending the classical Herz–Schur multipliers, making Herz–Schur multiplier techniques available to study a much larger class of C∗-algebras. Furthermore, we will also introduce and study generalised Schur multipliers, and derive links between these two notions which extend the classical results describing Herz–Schur multipliers in terms of Schur multipliers. This theory will be developed in as much generality as possible, with reference to the classical motivation. After introducing all the necessary concepts we begin the investigation by defining generalised Schur multipliers. The main result is a dilation type characterisation of these multipliers; we also show how such multipliers can be represented using HilbertC∗-modules. Next we introduce and study generalised Herz–Schur multipliers, first extending a classical result involving the representation theory of SU(2), before studying how such functions are related to our generalised Schur multipliers. We give a characterisation of generalised Herz–Schur multipliers as a certain class of the generalised Schur multipliers, and obtain a description of precisely which Schur multipliers belong to this class. Finally, we consider some ways in which the generalised multipliers can arise; firstly, from the classical multipliers which provide our motivation, secondly, from the Haagerup tensor product of a C∗-algebra with itself, and finally from positivity considerations. We show that our theory behaves well with respect to positivity and give conditions under which our multipliers are automatically positive in a natural sense.
12
Hook, James Louis. "Topics in dynamical systems." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/topics-in-dynamical-systems(427b5d98-197d-4b53-876e-a81142f72375).html.
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In this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These systems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures. In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue network impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue. In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically 'smooth' these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data.
13
Hua, Xinhou. "Dynamical systems and wavelets." Thesis, University of Ottawa (Canada), 2002. http://hdl.handle.net/10393/6143.
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The first part of this thesis is concerned with Bakers Conjecture (1984) which says that two permutable transcendental entire functions have the same Julia set. To this end, we shall exhibit that two permutable transcendental entire functions of a certain type have the same Julia set. So far, this is the best result to the conjecture. The second part relates to Newton's method to find zeros of functions. We shall look for the locations of the limits of the iterating sequence of the relaxed Newton function on its wandering domains. A relaxed Newton function with corresponding properties is constructed. The third part relates to the dynamics of ordinary differential equations and inverse problems. Given a target solution, we shall construct second-order differential equations with Legendre polynomial basis to approximate the target solution. An algorithm and numerical solutions are provided. Examples show that the approximations we have found are much better than the known results obtained by means of first-order differential equations. We shall also discuss approximation using a wavelet basis. MATLAB is used to compute the numerical results. In the fourth part, we deal with variational problems in signal and image processing. For a given signal or image represented by a function, we shall provide a good approximation to the function, which minimizes a given functional.
14
Dam, Albert Anton ten. "Unilaterally constrained dynamical systems." [S.l : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1997. http://irs.ub.rug.nl/ppn/159407869.
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15
Schinkel, Michael. "Nondeterministic hybrid dynamical systems." Thesis, University of Glasgow, 2002. http://theses.gla.ac.uk/1853/.
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This thesis is concerned with the analysis, control and identification of hybrid dynamical systems. The main focus is on a particular class of hybrid systems consisting of linear subsystems. The discrete dynamic, i.e., the change between subsystems, is unknown or nondeterministic and cannot be influenced, i.e. controlled, directly. However changes in the discrete dynamic can be detected immediately, such that the current dynamic (subsystem) is known. In order to motivate the study of hybrid systems and show the merits of hybrid control theory, an example is given. It is shown that real world systems like Anti Locking Brakes (ABS) are naturally modelled by such a class of linear hybrids systems. It is shown that purely continuous feedback is not suitable since it cannot achieve maximum braking performance. A hybrid control strategy, which overcomes this problem, is presented. For this class of linear hybrid system with unknown discrete dynamic, a framework for robust control is established. The analysis methodology developed gives a robustness radius such that the stability under parameter variations can be analysed. The controller synthesis procedure is illustrated in a practical example where the control for an active suspension of a car is designed. Optimal control for this class of hybrid system is introduced. It is shows how a control law is obtained which minimises a quadratic performance index. The synthesis procedure is stated in terms of a convex optimisation problem using linear matrix inequalities (LMI). The solution of the LMI not only returns the controller but also the performance bound. Since the proposed controller structures require knowledge of the continuous state, an observer design is proposed. It is shown that the estimation error converges quadratically while minimising the covariance of the estimation error. This is similar to the Kalman filter for discrete or continuous time systems. Further, we show that the synthesis of the observer can be cast into an LMI, which conveniently solves the synthesis problem.
16
Chan, N. "Dynamical systems in cosmology." Thesis, University College London (University of London), 2012. http://discovery.ucl.ac.uk/1348375/.
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In this PhD thesis, the role of dynamical systems in cosmology has been studied. Many systems and processes of cosmological interest can be modelled as dynamical systems. Motivated by the concept of hypothetical dark energy that is believed to be responsible for the recently discovered accelerated expansion of the universe, various dynamical dark energy models coupled to dark matter have been investigated using a dynamical systems approach. The models investigated include quintessence, three-form and phantom fields, interacting with dark matter in different forms. The properties of these models range from mathematically simple ones to those with better physical motivation and justification. It was often encountered that linear stability theory fails to reveal behaviour of the dynamical systems. As part of this PhD programme, other techniques such as application of the centre manifold theory, construction of Lyapunov functions were considered. Applications of these so-called methods of non-linear stability theory were applied to cosmological models. Aforementioned techniques are powerful tools that have direct applications not only in applied mathematics, theoretical physics and engineering, but also in finance, economics, theoretical immunology, neuroscience and many more. One of the main aims of this thesis is to bridge the gap between dynamical systems theory, an area of applied mathematics, and cosmology, an exciting area of physics that studies the universe as a whole.
17
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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Royals, Robert. "Arithmetic and dynamical systems." Thesis, University of East Anglia, 2015. https://ueaeprints.uea.ac.uk/57191/.
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In this thesis we look at a number of topics in the area of the interaction between dynamical systems and number theory. We look at two diophantine approximation problems in local �fields of positive characteristic, one a generalisation of the Khintchine{Groshev theorem, another a central limit theorem. We also prove a P�olya{Carlson dichotomy result for a large class of adelicly perturbed rational functions. In particular we prove that for a finite set of primes S that the power series f(z) generated by the Fibonacci series with all primes in S removed has a natural boundary.
19
Hayden, Kevin. "Modeling of dynamical systems /." abstract and full text PDF (UNR users only), 2007. http://0-gateway.proquest.com.innopac.library.unr.edu/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:1446796.
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Thesis (M.S.)--University of Nevada, Reno, 2007.<br>"May, 2007." Includes bibliographical references (leaves 128-129). Library also has microfilm. Ann Arbor, Mich. : ProQuest Information and Learning Company, [2008]. 1 microfilm reel ; 35 mm. Online version available on the World Wide Web.
20
Sun, Hongyan. "Coupled nonlinear dynamical systems." Morgantown, W. Va. : [West Virginia University Libraries], 2000. http://etd.wvu.edu/templates/showETD.cfm?recnum=1636.
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Thesis (Ph. D.)--West Virginia University, 2000.<br>Title from document title page. Document formatted into pages; contains xi, 113 p. : ill. (some col.). Includes abstract. Includes bibliographical references.
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Tse, Pak-hoi Isaac. "Dynamical systems theory and school change." Click to view the E-thesis via HKUTO, 2006. http://sunzi.lib.hku.hk/hkuto/record/B37626218.
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Tse, Pak-hoi Isaac, and 謝伯開. "Dynamical systems theory and school change." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2006. http://hub.hku.hk/bib/B37626218.
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Kuhlman, Christopher James. "Generalizations of Threshold Graph Dynamical Systems." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/76765.
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Dynamics of social processes in populations, such as the spread of emotions, influence, language, mass movements, and warfare (often referred to individually and collectively as contagions), are increasingly studied because of their social, political, and economic impacts. Discrete dynamical systems (discrete in time and discrete in agent states) are often used to quantify contagion propagation in populations that are cast as graphs, where vertices represent agents and edges represent agent interactions. We refer to such formulations as graph dynamical systems. For social applications, threshold models are used extensively for agent state transition rules (i.e., for vertex functions). In its simplest form, each agent can be in one of two states (state 0 (1) means that an agent does not (does) possess a contagion), and an agent contracts a contagion if at least a threshold number of its distance-1 neighbors already possess it. The transition to state 0 is not permitted. In this study, we extend threshold models in three ways. First, we allow transitions to states 0 and 1, and we study the long-term dynamics of these bithreshold systems, wherein there are two distinct thresholds for each vertex; one governing each of the transitions to states 0 and 1. Second, we extend the model from a binary vertex state set to an arbitrary number r of states, and allow transitions between every pair of states. Third, we analyze a recent hierarchical model from the literature where inputs to vertex functions take into account subgraphs induced on the distance-1 neighbors of a vertex. We state, prove, and analyze conditions characterizing long-term dynamics of all of these models.<br>Master of Science
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Nersesov, Sergey G. "Nonlinear Impulsive and Hybrid Dynamical Systems." Diss., Georgia Institute of Technology, 2005. http://hdl.handle.net/1853/7147.
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Modern complex dynamical systems typically possess a
multiechelon hierarchical hybrid structure characterized by
continuous-time dynamics at the lower-level units and logical
decision-making units at the higher-level of hierarchy. Hybrid
dynamical systems involve an interacting countable collection of
dynamical systems defined on subregions of the partitioned state
space. Thus, in addition to traditional control systems, hybrid
control systems involve supervising controllers which serve to
coordinate the (sometimes competing) actions of the lower-level
controllers. A subclass of hybrid dynamical systems are impulsive
dynamical systems which consist of three elements, namely, a
continuous-time differential equation, a difference equation, and
a criterion for determining when the states of the system are to
be reset. One of the main topics of this dissertation is the
development of stability analysis and control design for impulsive
dynamical systems. Specifically, we generalize Poincare's
theorem to dynamical systems possessing left-continuous flows to
address the stability of limit cycles and periodic orbits of
left-continuous, hybrid, and impulsive dynamical systems. For
nonlinear impulsive dynamical systems, we present partial
stability results, that is, stability with respect to part of the
system's state. Furthermore, we develop adaptive control framework
for general class of impulsive systems as well as energy-based
control framework for hybrid port-controlled Hamiltonian systems.
Extensions of stability theory for impulsive dynamical systems
with respect to the nonnegative orthant of the state space are
also addressed in this dissertation. Furthermore, we design
optimal output feedback controllers for set-point regulation of
linear nonnegative dynamical systems. Another main topic that has
been addressed in this research is the stability analysis of
large-scale dynamical systems. Specifically, we extend the theory
of vector Lyapunov functions by constructing a generalized
comparison system whose vector field can be a function of the
comparison system states as well as the nonlinear dynamical system
states. Furthermore, we present a generalized convergence result
which, in the case of a scalar comparison system, specializes to
the classical Krasovskii-LaSalle invariant set theorem. Moreover,
we develop vector dissipativity theory for large-scale dynamical
systems based on vector storage functions and vector supply rates.
Finally, using a large-scale dynamical systems perspective, we
develop a system-theoretic foundation for thermodynamics.
Specifically, using compartmental dynamical system energy flow
models, we place the universal energy conservation, energy
equipartition, temperature equipartition, and entropy
nonconservation laws of thermodynamics on a system-theoretic
basis.
25
Shadden, Shawn Christopher Marsden Jerrold E. "A dynamical systems approach to unsteady systems /." Diss., Pasadena, Calif. : Caltech, 2006. http://resolver.caltech.edu/CaltechETD:etd-05122006-083011.
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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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Gil, Gibin. "Hybrid Numerical Integration Scheme for Highly Oscillatory Dynamical Systems." Diss., The University of Arizona, 2013. http://hdl.handle.net/10150/306771.
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Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration algorithms. A system is considered to be highly oscillatory if it contains a fast solution that varies regularly about a slow solution. As for multibody systems, stiff force elements and contacts between bodies can make a system highly oscillatory. Standard explicit numerical integration methods should take a very small step size to satisfy the absolute stability condition for all eigenvalues of the system and the computational cost is dictated by the fast solution. In this research, a new hybrid integration scheme is proposed, in which the local linearization method is combined with a conventional integration method such as the fourth-order Runge-Kutta. In this approach, the system is partitioned into fast and slow subsystems. Then, the two subsystems are transformed into a reduced and a boundary-layer system using the singular perturbation theory. The reduced system is solved by the fourth-order Runge-Kutta method while the boundary-layer system is solved by the local linearization method. This new hybrid scheme can handle the coupling between the fast and the slow subsystems efficiently. Unlike other multi-rate or multi-method schemes, extrapolation or interpolation process is not required to deal with the coupling between subsystems. Most of the coupling effect can be accounted for by the reduced (or quasi-steady-state) system while the minor transient effect is taken into consideration by averaging. In this research, the absolute stability region for this hybrid scheme is derived and it is shown that the absolute stability region is almost independent of the fast variables. Thus, the selection of the step size is not dictated by the fast solution when a highly oscillatory system is solved, in turn, the computational efficiency can be improved. The advantage of the proposed hybrid scheme is validated through several dynamic simulations of a vehicle system including a flexible tire model. The results reveal that the hybrid scheme can reduce the computation time of the vehicle dynamic simulation significantly while attaining comparable accuracy.
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Zhao, Zhenyuan. "Dynamical Grouping in Complex Systems." Scholarly Repository, 2010. http://scholarlyrepository.miami.edu/oa_dissertations/498.
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Quantifying the behavior of complex systems arguably presents the common ¡°hard¡±problem across the physical, biological, social, economic sciences. Individual-based or agent-based models have proved useful in a variety of different real world systems: from the physical, biological, medical domains through to social and even financial domains. There are many different models in each of these fields, each with their own particular assumptions, strengths and weaknesses for particular application areas. However, there is a lack of minimal model analysis in which both numerical and analytic results can be obtained, and hence allowing different application domains to be analyzed on a common footing. This thesis focuses on a few simple, yet highly non-trivial, minimal models of a population of interacting objects (so-called agents) featuring internal dynamical grouping. In addition to analyzing these models, I apply them to a number of distinct real world systems. Both the numerical and analytical results suggest that these simple models could be key factors in explaining the overall collective behavior and emergent properties in a wide range of real world complex systems. In particular, I study variants of a particular model (called the EZ model) in order to explain the attrition time in modern conflicts, and the evolution of contagion phenomena in such a dynamically evolving population. I also study and explain the empirical data obtained for online guilds and offline gangs, leading to a team-based model which captures the common quantitative features of the data. I then move on to develop a resource competition model (i.e. the so-called El Farol model) and apply it to the carbon emissions market, mapping the different market factors into model parameters which enable me to explore the potential market behaviors under a variety of scenarios.
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Badar, Muhammad. "Dynamical Systems Over Finite Groups." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17948.
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In this thesis, the dynamical system is used as a function on afinite group, to show how states change. We investigate the'numberof cycles' and 'length of cycle' under finite groups. Using grouptheory, fixed point, periodic points and some examples, formulas tofind 'number of cycles' and 'length of cycle' are derived. Theexamples used are on finite cyclic group Z_6 with respectto binary operation '+'. Generalization using finite groups ismade. At the end, I compared the dynamical system over finite cyclic groups with the finite non-cyclic groups and then prove the general formulas to find 'number of cycles' and 'length of cycle' for both cyclic and non-cyclic groups.
30
Newman, Julian. "Synchronisation in random dynamical systems." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/39569.
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In this thesis, we develop a deeper and much more extensive theory of synchronisation of trajectories of random dynamical systems (RDS) than currently exists. In particular, focusing on random dynamical systems with memoryless noise, we achieve two main goals: Firstly, we demonstrate that the notion of "statistical equilibria" is purely a property of the measurable dynamics of a RDS on a standard Borel space; and yet, within such statistical equilibria is "encoded" the phenomenon of noise-induced synchronisation (which may then be observed in *any* compatible metric on the phase space). Secondly, we provide new, widely applicable criteria for synchronisation in RDS, considerably improving upon some of the existing criteria for synchronisation.
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Hendtlass, Matthew Ralph John. "Aspects of Constructive Dynamical Systems." Thesis, University of Canterbury. Mathematics and Statistics, 2009. http://hdl.handle.net/10092/2724.
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We give a Bishop-style constructive analysis of the statement that a continuous homomorphism from the real line onto a compact metric abelian group is periodic; constructive versions of this statement and its contrapositive are given. It is shown that the existence of a minimal period in general is not derivable, but the minimal period is derivable under a simple geometric condition when the group is contained in two dimensional Euclidean space. A number of results about one-one and injective mappings are proved en route to our main theorems. A few Brouwerian examples show that some of our results are the best possible in a constructive framework.
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Ozik, Jonathan. "Evolution of discrete dynamical systems." College Park, Md. : University of Maryland, 2005. http://hdl.handle.net/1903/2351.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2005.<br>Thesis research directed by: Physics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Ertem, Turker. "Asymptotic Integration Of Dynamical Systems." Phd thesis, METU, 2013. http://etd.lib.metu.edu.tr/upload/12615405/index.pdf.
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In almost all works in the literature there are several results showing asymptotic relationships between the solutions of
x&prime<br>&prime<br>= f (t, x) (0.1)
and the solutions 1 and t of x&prime<br>&prime<br>= 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b &isin<br>R has been obtained.
In this thesis we investigate in a systematic way the asymptotic behavior as t &rarr<br>&infin<br>of solutions of a class of differential equations of the form
(p(t)x&prime<br>)&prime<br>+ q(t)x = f (t, x), t &ge<br>t_0 (0.2)
and
(p(t)x&prime<br>)&prime<br>+ q(t)x = g(t, x, x&prime<br>), t &ge<br>t_0 (0.3)
by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation
(p(t)x&prime<br>)&prime<br>+ q(t)x = 0, t &ge<br>t_0. (0.4)
Here, t_0 &ge<br>0 is a real number, p &isin<br>C([t_0,&infin<br>), (0,&infin<br>)), q &isin<br>C([t_0,&infin<br>),R), f &isin<br>C([t_0,&infin<br>) ×<br>R,R) and g &isin<br>C([t0,&infin<br>) ×<br>R ×<br>R,R).
Our argument is based on the idea of writing the solution of x&prime<br>&prime<br>= 0 in terms of principal and nonprincipal solutions as x(t) = av(t) + bu(t), where v(t) = t and u(t) = 1.
In the proofs, Banach and Schauder&rsquo<br>s fixed point theorems are used. The compactness of the operator is obtained by employing the compactness criteria of Riesz and Avramescu.
The thesis consists of three chapters. Chapter 1 is introductory and provides statement of the problem, literature review, and basic definitions and theorems.
In Chapter 2 first we deal with some asymptotic relationships between the solutions of (0.2) and the principal u(t) and nonprincipal v(t) solutions of (0.4). Then we present existence of a monotone positive solution of (0.3) with prescribed asimptotic behavior.
In Chapter 3 we introduce the existence of solution of a singular boundary value problem to the Equation (0.2).
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Johnson, M. E. "Bifurcations in lattice dynamical systems." Thesis, University of Cambridge, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.605623.
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In this thesis I consider bifurcations in lattice dynamical systems, primarily coupled map lattices and lattice differential equations. First I explain how certain coupled map lattices can be considered as cellular automata, on all or part of each of state- or parameter-space, and outline what can be gained from such a consideration. Then I consider bifurcations of fixed points in locally bistable coupled map lattices, introducing various analytical techniques for the study of piecewise-linear lattice dynamical systems, and showing how numerical techniques allow smooth systems to be investigated also. In particular, bifurcation diagrams and bifurcation sets for "kink" and "bump" fixed points are constructed, and these reveal that there will be small regions of parameter space in which no stable bump fixed points exist, though many such fixed points exist on each side of these regions. Strange behaviour occurs in such regions, with very long transients being observed. Bifurcations from homogenous fixed points of lattice dynamical systems are then investigated using the methods of equivalent bifurcation theory. As well as locating such bifurcations, I obtain information regarding branching directions, numbers of branches, and branch stabilities. The effect of the local dynamical units having odd symmetry is discussed. Some aspects of the theory of infinite-dimensional lattice dynamical systems are then considered. I discuss the connection between fixed points of these systems and orbits of a certain area-preserving map of the plane, and extend some earlier results regarding the symbolic dynamics of this map. I then introduce a shadowing-based technique which allows us to infer the existence of certain fixed points on the infinite lattice using our knowledge of similar fixed points on finite lattices.
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Bernhard, Michael A. "Introduction to chaotic dynamical systems." Thesis, Monterey, California. Naval Postgraduate School, 1992. http://hdl.handle.net/10945/23708.
Full textAbstract:
The emerging discipline known as "chaos theory" is a relatively new field of study with a diverse range of applications (economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for "chaos" as it applies to gen- eral dynamical systems. Various approaches range from topological methods of a qualitative description, to physical notions of randomness, information, and entropy in crgodic theory, to the development of computational definitions and algorithms designed to obtain quantitative information.
This thesis develops some of the current definitions and discusses several quantita- tive measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems it serves as a complement to the work done by Philip Beaver [Ref. 1], which details chaotic dynamics for discrete systems.
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Clewlow, Les. "Cellular automata and dynamical systems." Thesis, University of Warwick, 1989. http://wrap.warwick.ac.uk/4233/.
Full textAbstract:
In this thesis we investigate the theoretical nature of the mathematical structures termed cellular automata. Chapter 1: Reviews the origin and history of cellular automata in order to place the current work into context. Chapter 2: Develops a cellular automata framework which contains the main aspects of cellular automata structure which have appeared in the literature. We present a scheme for specifying the cellular automata rules for this general model and present six examples of cellular automata within the model. Chapter 3: Here we develop a statistical mechanical model of cellular automata behaviour. We consider the relationship between variations within the model and their relationship to dynamical systems. We obtain results on the variance of the state changes, scaling of the cellular automata lattice, the equivalence of noise, spatial mixing of the lattice states and entropy, synchronous and asynchronous cellular automata and the equivalence of the rule probability and the time step of a discrete approximation to a dynamical system. Chapter 4: This contains an empirical comparison of cellular automata within our general framework and the statistical mechanical model. We obtain results on the transition from limit cycle to limit point behaviour as the rule probabilities are decreased. We also discuss failures of the statistical mechanical model due to failure of the assumptions behind it. Chapter 5: Here a practical application of the preceding work to population genetics is presented. We study this in the context of some established population models and show it may be most useful in the field of epidemiology. Further generalisations of the statistical mechanical and cellular automata models allow the modelling of more complex population models and mobile populations of organisms. Chapter 6: Reviews the results obtained in the context of the open questions introduced in Chapter 1. We also consider further questions this work raises and make some general comments on how these may apply to related fields.
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Garira, Winston. "Synchronisation of coupled dynamical systems." Thesis, University of London, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.399495.
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Bandtlow, Oscar F. "Spectral analysis of dynamical systems." Thesis, University of Cambridge, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396095.
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Campanella, Giammarco. "Dynamical aspects of exoplanetary systems." Thesis, Queen Mary, University of London, 2013. http://qmro.qmul.ac.uk/xmlui/handle/123456789/8374.
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The detection of more than 130 multiple planet systems makes it necessary to interpret a broader range of properties than are shown by our Solar system. This thesis covers aspects linked to the proliferation in recent years of multiple extrasolar planet systems. A narrow observational window, only partially covering the longest orbital period, can lead to solutions representing unrealistic scenarios. The best-fit solution for the three-planet extrasolar system of HD 181433 describes a highly unstable configuration. Taking into account the dynamical stability as an additional observable while interpreting the RV data, I have analysed the phase space in the neighbourhood of the statistical best-fit. The two giant companions are found to be locked in the 5:2 MMR in the stable best-fit model. I have analysed the dynamics of the system HD 181433 by assessing different scenarios that may explain the origin of these eccentric orbits, with particular focus on the innermost body. A scenario is considered in which the system previously contained an additional giant planet that was ejected during a period of dynamical instability among the planets. Also considered is a scenario in which the spin-down of the central star causes the system to pass through secular resonance. In its simplest form this latter scenario fails to produce the system observed. If additional short-period low mass planets are present in the system, I find that mutual scattering can release planet b from the secular resonance, leading to a system with orbital parameters similar to those observed today. Finally, I have studied the evolution of low mass planets interacting with a gas-giant planet embedded in a gaseous disc. The transit timing method allows the detection of non-transiting planets through their gravitational perturbations. I have investigated the detectability of low mass planets neighbouring short-period giants after protoplanetary disc dispersal.
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Joyner, Sheldon T. "On non-archimedean dynamical systems." Thesis, Stellenbosch : Stellenbosch University, 2000. http://hdl.handle.net/10019.1/51861.
Full textAbstract:
Thesis (MSc) -- University of Stellenbosch, 2000.<br>ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map
cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points
and subsets of X are classified according to their behaviour under iteration. These subsets
include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic
points, and many interesting questions arise in the study of their properties.
Such questions have been extensively studied in the case of complex dynamics, but much
recent work has focussed on non-archimedean dynamical systems, when X is projective
space over some field equipped with a non-archimedean metric. This work has uncovered
many parallels to complex dynamics alongside more striking differences.
In this thesis, various aspects of the theory of non-archimedean dynamics are presented,
with particular reference to JULIA and FATOU sets and the relationship between good
reduction of a map and the empty JULIA set. We also discuss questions of the finiteness
of the sets of periodic points in special contexts.<br>AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X
vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op
die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling.
Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op
hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings
van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van
hierdie versamelings kom na vore.
In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse
werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is
oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie
ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys.
In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese
dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en
die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae
oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook
aangebied.
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Frisk, Martin. "Synchronization in chaotic dynamical systems." Thesis, Uppsala universitet, Tillämpad matematik och statistik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-287624.
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Liu, Zheng. "Dynamical systems and random perturbations." Diss., The University of Arizona, 1993. http://hdl.handle.net/10150/186232.
Full textAbstract:
This dissertation consists of two independent parts. In the first part we study the ergodic theory of surface endomorphisms. We consider non-uniformly expanding maps with generic singularities, and prove that the Pesin formula holds, which is to say that entropy is equal to the sum of the positive Lyapunov exponents if and only if the invariant probability measure in question is absolutely continuous with respect to Lebesgue measure. In the second part we study the small random perturbations of the Feigenbaum map related to the fixed point of Feigenbaum's renormalization operator for unimodal maps of the interval. We give a rigorous analysis of the changes in the geometry of the noisy attractor as noise level varies.
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Carracedo, Rodriguez Andrea. "Approximation of Parametric Dynamical Systems." Diss., Virginia Tech, 2020. http://hdl.handle.net/10919/99895.
Full textAbstract:
Dynamical systems are widely used to model physical phenomena and, in many cases, these
physical phenomena are parameter dependent. In this thesis we investigate three prominent
problems related to the simulation of parametric dynamical systems and develop the analysis
and computational framework to solve each of them.
In many cases we have access to data resulting from simulations of a parametric dynamical
system for which an explicit description may not be available. We introduce the parametric
AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric
dynamical system from its input/output measurements, in the form of transfer function
evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational
approximation of nonparametric systems, to the parametric case. We develop p-AAA for both
scalar and matrix-valued data and study the impact of parameter scaling. Even though we
present p-AAA with parametric dynamical systems in mind, the ideas can be applied to
parametric stationary problems as well, and we include such examples.
The solution of a dynamical system can often be expressed in terms of an eigenvalue problem
(EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A
common approach to solving (nonparametric) nonlinear EVPs is to approximate them with
a rational EVP and then to linearize this approximation. An existing algorithm can then be
applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully
applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define
a corresponding linearization that fits the format of the compact rational Krylov (CORK)
algorithm for the approximation of eigenvalues.
The simulation of dynamical systems may be costly, since the need for accuracy may yield a
system of very large dimension. This cost is magnified in the case of parametric dynamical
systems, since one may be interested in simulations for many parameter values. Interpolatory
model order reduction (MOR) tackles this problem by creating a surrogate model that
interpolates the original, is of much smaller dimension, and captures the dynamics of the
quantities of interest well. We generalize interpolatory projection MOR methods from parametric
linear to parametric bilinear systems. We provide necessary subspace conditions to
guarantee interpolation of the subsystems and their first and second derivatives, including
the parameter gradients and Hessians.
Throughout the dissertation, the analysis is illustrated via various benchmark numerical
examples.<br>Doctor of Philosophy<br>Simulation of mathematical models plays an important role in the development of science.
There is a wide range of models and approaches that depend on the information available
and the goal of the problem. In this dissertation we focus on three problems whose solution
depends on parameters and for which we have either data resulting from simulations of the
model or a explicit structure that describes the model. First, for the case when only data are
available, we develop an algorithm that builds a data-driven approximation that is then easy
to reevaluate. Second, we embed our algorithm in an already developed framework for the
solution of a specific kind of model structure: nonlinear eigenvalue problems. Third, given
a model with a specific nonlinear structure, we develop a method to build a model with the
same structure, smaller dimension (for faster computation), and that provides an accurate
approximation of the original model.
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Grimm, Alexander Rudolf. "Taming of Complex Dynamical Systems." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/24775.
Full textAbstract:
The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the problem of proving global existence and uniqueness is more difficult and fails even for some very simple ordinary differential equations. It is still not known if the 3D Navier-Stokes equation have global unique solutions and this open problem is one of the Millennium Prize Problems. However, many of these mathematical models are extremely useful in the understanding of complex physical systems. For years people have considered methods for modifying these equations in order to obtain models that still capture the observed fundamental physics, but for which one can rigorously establish global results. In this thesis we focus on a taming method to achieve this goal and apply taming to modeling and numerical problems. The method is also applied to a class of nonlinear differential equations with conservative nonlinearities and to Burgers’ Equation with Neumann boundary conditions. Numerical results are presented to illustrate the ideas.<br>Master of Science
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Mcnitt, Joseph Andrew. "Stability in Graph Dynamical Systems." Thesis, Virginia Tech, 2018. http://hdl.handle.net/10919/83604.
Full textAbstract:
The underlying mathematical model of many simulation models is graph dynamical systems (GDS). This dynamical system, its implementation, and analyses on each will be the focus of this paper. When using a simulation model to answer a research question, it is important to describe this underlying mathematical model in which we are operating for verification and validation. In this paper we discuss analyses commonly used in simulation models. These include sensitivity analyses and uncertainty quantification, which provide motivation for stability and structure-to-function research in GDS. We review various results in these areas, which contribute toward validation and computationally tractable analyses of our simulation model. We then present two new areas of research - stability of transient structure with respect to update order permutations, and an application of GDS in which a time-varying generalized cellular automata is implemented as a simulation model.<br>Master of Science
46
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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Bhagavatula, Ravi S. "Topics in extended dynamical systems /." The Ohio State University, 1994. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487856076416283.
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Kupsa, Michal. "Return times in dynamical systems." Toulon, 2005. http://www.theses.fr/2005TOUL0009.
Full textAbstract:
Dans cette thèse, deux aspects asymptotiques des temps de retour et d'entrée sont étudiés: les taux locaux de temps de retour, et les lois limites des k-ièmes temps de retour et d'entrée. Dans le cadre des "shifts" Sturmiens, des formules permettant de calculer les taux locaux de temps de retour sont développées. La classe des lois limites de premier temps d'entrée est décrite explicitement. Nous y prouvons que la classe des lois limites des k-ièmes temps de retour est la même que celle des premiers temps de retour, caractérisée par Y. Lacroix. Enfin, nous y exhibons un lien entre les lois limites des k-ièmes temps de retour et d'entrée<br>Several statistics of hitting and return times in dynamical systems are investigated in this thesis. It concerns local return rates and the limit laws of k-th return and hitting times. Formulas to compute the local return rates in Sturmian shifts are developed. The class of all limit laws of the first hitting times is described. The class of all limit laws of the k-th return times is shown to be the same as the class of all limit laws of the first return times, characterized by Lacroix. Last but not least, we exhibit a link between k-limit laws of return and hitting times
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Stergiopoulou, Aikaterini. "Dynamical Stability of Planetary Systems." Thesis, Uppsala universitet, Institutionen för fysik och astronomi, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-323006.
Full textAbstract:
The study of dynamical stability in planetary systems has become possible during the last few decades due to the development of numerical methods for long-term integrations of N-body systems. Since the 90’s the number of exoplanet detections has been increased significantly, making the simulations of other real planetary systems besides the Solar System feasible. One of the exciting new-found worlds is the system Kepler-11. Six planets which are located very close to each other orbit a solar-type star. In this project we first investigate the behavior of Kepler-11 when we change some of the initial conditions of the outermost planet of the system and then we approximate the Red Giant phase of solar-type stars in order to see how the planetary orbits are altered. For the first part we run three series of simulations (groups A,B,C). Each group has a different value for the mean density of planet Kepler-11g (1.0,1.5,2.0 g/cm 3 ). We run simulations for 36 different combinations of mass and eccentricity of planet Kepler-11g for each group. In nine configurations all six planets of the system continue to orbit the star until the end of the simulations. These nine stable configurations of Kepler-11 are used in the second part where we implement a constant mass-loss rate for the star which results in 30% mass loss after 30 million years, trying to approximate that way the mass loss of solar-type stars in Red Giant Branch. We also run nine simulations of a hypothetical system consisting only of the Sun, Earth and Jupiter where we implement the constant mass-loss rate to the Sun. In the Kepler-11 system, the orbits of planets Kepler-11g and Kepler-11e change by ∼45% and ∼54% respectively, after 30 million years, due to the mass loss of the star, while in the hypothetical planetary system the orbits of the two planets change by ∼43%. The study of orbits and how they move outward during the Post-Main Sequence evolution of stars is essential for our understanding of the existence of a Habitable Zone, not just around stars in Main-Sequence phase, but also around stars in late stages of their evolution.
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Dundar, Veli Ufuktepe Ünal. "Dynamical Systems on Time Scales/." [s.l.]: [s.n.], 2007. http://library.iyte.edu.tr/tezler/master/matematik/T000648.pdf.
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