Thèses : « Nonlinear systems of equations »

1

Hadad, Yaron. « Integrable Nonlinear Relativistic Equations ». Diss., The University of Arizona, 2013. http://hdl.handle.net/10150/293490.

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This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied. It is also proven that it is equivalent to a novel equation that poses a fascinating similarity to the Sinh-Gordon equation. For the 1+1 Einstein equation the Belinski-Zakharov transformation is explored. It is used to derive explicit formula for N gravitational solitons on arbitrary diagonal background. In particular, the method is used to derive gravitational solitons on the Einstein-Rosen background. The similarities and differences between the attributes of the solitons of the symmetric Chiral field equation and those of the 1+1 Einstein equation are emphasized, and their origin is pointed out. For the 1+2 Einstein equation, new equations describing diagonal metrics are derived and their compatibility is proven. Different gravitational waves are studied that naturally extend the class of Bondi-Pirani-Robinson waves. It is further shown that the Bondi-Pirani-Robinson waves are stable with respect to perturbations of the spacetime. Their stability is closely related to the stability of the Schwarzschild black hole and the relation between the two allows to conjecture about the stability of a wide range of gravitational phenomena. Lastly, a new set of equations that describe weak gravitational waves is derived. This new system of equations is closely and fundamentally connected with the nonlinear Schrödinger equation and can be properly called the nonlinear Schrödinger-Einstein equations. A few preliminary solutions are constructed.

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Zerihun, Tadesse G. « Nonlinear Techniques for Stochastic Systems of Differential Equations ». Scholar Commons, 2013. http://scholarcommons.usf.edu/etd/4970.

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Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential equations. A complex nonlinear nonstationary stochastic system of differential equations are decomposed into nonlinear systems of stochastic perturbed and unperturbed differential equations. Using this type of decomposition, the fundamental properties of solutions of nonlinear stochastic unperturbed systems of differential equations are investigated(1). The fundamental properties are used to find the representation of solution process of nonlinear stochastic complex perturbed system in terms of solution process of nonlinear stochastic unperturbed system(2). Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we establish generalized variation of constants formula for solution process of perturbed stochastic system of differential equations(3). Results regarding deviation of solution of perturbed system with respect to solution of nominal system of stochastic differential equations are developed(4). The obtained results are used to study the qualitative properties of perturbed stochastic system of differential equations(5). Examples are given to illustrate the usefulness of the results. Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we establish generalized variational comparison theorems in the context of stochastic and deterministic differential for solution processes of perturbed stochastic system of differential equations(6). Results regarding deviation of solutions with respect to nominal stochastic system are also developed(7). The obtained results are used to study the qualitative properties of perturbed stochastic system(8). Examples are given to illustrate the usefulness of the results. A simple dynamical model of the effect of radiant flux density and CO_2 concentration on the rate of photosynthesis in light, dark and enzyme reactions are analyzed(9). The coupled system of dynamic equations are solved numerically for some values of rate constant and radiant flux density. We used Matlab to solve the system numerically. Moreover, with the assumption that dynamic model of CO_2 concentration is studied.

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3

Jaschke, Leonhard. « Preconditioned Arnoldi methods for systems of nonlinear equations / ». Paris (121 Av. des Champs-Élysées, 75008) : Wiku, 2004. http://catalogue.bnf.fr/ark:/12148/cb391991990.

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4

Foley, Dawn Christine. « Applications of State space realization of nonlinear input/output difference equations ». Thesis, Georgia Institute of Technology, 1999. http://hdl.handle.net/1853/16818.

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5

Van, der Walt Jan Harm. « Generalized solutions of systems of nonlinear partial differential equations ». Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-05242009-122628.

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6

Reichelt, Sina. « Two-scale homogenization of systems of nonlinear parabolic equations ». Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2015. http://dx.doi.org/10.18452/17385.

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Ziel dieser Arbeit ist es zwei verschiedene Klassen von Systemen nichtlinearer parabolischer Gleichungen zu homogenisieren, und zwar Reaktions-Diffusions-Systeme mit verschiedenen Diffusionslängenskalen und Gleichungen vom Typ Cahn-Hilliard. Wir betrachten parabolische Gleichungen mit periodischen Koeffizienten, wobei die Periode dem Verhältnis der charakteristischen mikroskopischen zu der makroskopische Längenskala entspricht. Unser Ziel ist es, effektive Gleichungen rigoros herzuleiten, um die betrachteten Systeme besser zu verstehen und den Simulationsaufwand zu minimieren. Wir suchen also einen Konvergenzbegriff, mit dem die Lösung des Ausgangsmodells im Limes der Periode gegen Null gegen die Lösung des effektiven Modells konvergiert. Um die periodische Mikrostruktur und die verschiedenen Diffusivitäten zu erfassen, verwenden wir die Zwei-Skalen Konvergenz mittels periodischer Auffaltung. Der erste Teil der Arbeit handelt von Reaktions-Diffusions-Systemen, in denen einige Spezies mit der charakteristischen Diffusionslänge der makroskopischen Skala und andere mit der mikroskopischen diffundieren. Die verschiedenen Diffusivitäten führen zu einem Verlust der Kompaktheit, sodass wir nicht direkt den Grenzwert der nichtlinearen Terme bestimmen können. Wir beweisen mittels starker Zwei-Skalen Konvergenz, dass das effektive Modell ein zwei-skaliges Modell ist, welches von der makroskopischen und der mikroskopischen Skale abhängt. Unsere Methode erlaubt es uns, explizite Raten für die Konvergenz der Lösungen zu bestimmen. Im zweiten Teil betrachten wir Gleichungen vom Typ Cahn-Hilliard, welche ortsabhängige Mobilitätskoeffizienten und allgemeine Potentiale beinhalten. Wir beweisen evolutionäre Gamma-Konvergenz der zugehörigen Gradientensysteme basierend auf der Gamma-Konvergenz der Energien und der Dissipationspotentiale.
The aim of this thesis is to derive homogenization results for two different types of systems of nonlinear parabolic equations, namely reaction-diffusion systems involving different diffusion length scales and Cahn-Hilliard-type equations. The coefficient functions of the considered parabolic equations are periodically oscillating with a period which is proportional to the ratio between the charactersitic microscopic and macroscopic length scales. In view of greater structural insight and less computational effort, it is our aim to rigorously derive effective equations as the period tends to zero such that solutions of the original model converge to solutions of the effective model. To account for the periodic microstructure as well as for the different diffusion length scales, we employ the method of two-scale convergence via periodic unfolding. In the first part of the thesis, we consider reaction-diffusion systems, where for some species the diffusion length scale is of order of the macroscopic length scale and for other species it is of order of the microscopic one. Based on the notion of strong two-scale convergence, we prove that the effective model is a two-scale reaction-diffusion system depending on the macroscopic and the microscopic scale. Our approach supplies explicit rates for the convergence of the solution. In the second part, we consider Cahn-Hilliard-type equations with position-dependent mobilities and general potentials. It is well-known that the classical Cahn-Hilliard equation admits a gradient structure. Based on the Gamma-convergence of the energies and the dissipation potentials, we prove evolutionary Gamma-convergence, for the associated gradient system such that we obtain in the limit of vanishing periods a Cahn-Hilliard equation with homogenized coefficients.

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7

Alam, Md Shafiful. « Iterative Methods to Solve Systems of Nonlinear Algebraic Equations ». TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2305.

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Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different acceleration techniques are discussed with analysis of the asymptotic behavior of the iterates. Analogies between single variable and multivariable problems are detailed. We also explore some interesting phenomena while analyzing Newton's method for complex variables.

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8

Handel, Andreas. « Limits of Localized Control in Extended Nonlinear Systems ». Diss., Georgia Institute of Technology, 2004. http://hdl.handle.net/1853/5025.

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We investigate the limits of localized linear control in spatially extended, nonlinear systems. Spatially extended, nonlinear systems can be found in virtually every field of engineering and science. An important category of such systems are fluid flows. Fluid flows play an important role in many commercial applications, for instance in the chemical, pharmaceutical and food-processing industries. Other important fluid flows include air- or water flows around cars, planes or ships. In all these systems, it is highly desirable to control the flow of the respective fluid. For instance control of the air flow around an airplane or car leads to better fuel-economy and reduced noise production. Usually, it is impossible to apply control everywhere. Consider an airplane: It would not be feasibly to cover the whole body of the plane with control units. Instead, one can place the control units at localized regions, such as points along the edge of the wings, spaced as far apart from each other as possible. These considerations lead to an important question: For a given system, what is the minimum number of localized controllers that still ensures successful control? Too few controllers will not achieve control, while using too many leads to unnecessary expenses and wastes resources. To answer this question, we study localized control in a class of model equations. These model equations are good representations of many real fluid flows. Using these equations, we show how one can design localized control that renders the system stable. We study the properties of the control and derive several expressions that allow us to determine the limits of successful control. We show how the number of controllers that are needed for successful control depends on the size and type of the system, as well as the way control is implemented. We find that especially the nonlinearities and the amount of noise present in the system play a crucial role. This analysis allows us to determine under which circumstances a given number of controllers can successfully stabilize a given system.

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9

Twiton, Michael. « Analysis of Singular Solutions of Certain Painlevé Equations ». Thesis, The University of Sydney, 2018. http://hdl.handle.net/2123/18206.

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The six Painlevé equations can be described as the boundary between the non- integrable- and the trivially integrable-systems. Ever since their discovery they have found numerous applications in mathematics and physics. The solutions of the Painlevé equations are, in most cases, highly transcendental and hence cannot be expressed in closed form. Asymptotic methods do better, and can establish the behaviour of some of the solutions of the Painlevé equations in the neighbourhood of a singularity, such as the point at infinity. Although the quantitative nature of these neighbourhoods is not initially implied from the asymptotic analysis, some regularity results exist for some of the Painlevé equations. In this research, we will present such results for some of the remaining Painlevé equations. In particular, we will provide concrete estimates of the intervals of analyticity of a one-parameter family of solutions of the second Painlevé equation, and estimate the domain of analyticity of a “triply-truncated” solution of the fourth Painlevé equation. In addition we will also deduce the existence of solutions with particular asymptotic behaviour for the discrete Painlevé equations, which are discrete integrable nonlinear systems.

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10

Liu, Weian, Yin Yang et Gang Lu. « Viscosity solutions of fully nonlinear parabolic systems ». Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2621/.

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In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].

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11

Floater, Michael S. « Blow-up of solutions to nonlinear parabolic equations and systems ». Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.235037.

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12

Craciun, Gheorghe. « Systems of nonlinear equations deriving from complex chemical reaction networks / ». The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486463321624709.

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13

Zhelezov, Gleb, et Gleb Zhelezov. « Coalescing Particle Systems and Applications to Nonlinear Fokker-Planck Equations ». Diss., The University of Arizona, 2017. http://hdl.handle.net/10150/624562.

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We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

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14

Wu, Xiaoming. « Partial differential equations with applications to wave propagation ». Thesis, University of Newcastle Upon Tyne, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239573.

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15

Leiva, Hugo. « Skew-product semiflows and time-dependent dynamical systems ». Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/29912.

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Rejoub, Riad A. « Projective and non-projective systems of first order nonlinear differential equations ». Scholarly Commons, 1992. https://scholarlycommons.pacific.edu/uop_etds/2228.

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It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait. In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas] The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.

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See, Chong Wee Simon. « Numerical methods for the simulation of dynamic discontinuous systems ». Thesis, University of Salford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358276.

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Klug, Andreas. « Affine-scaling methods for nonlinear minimization problems and nonlinear systems of equations with bound constraints ». Doctoral thesis, [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=981743978.

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Mulvey, Joseph Anthony. « Symmetry methods for integrable systems ». Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5379/.

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This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.

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Gurevich, Svetlana V. « Lateral self-organization in nonlinear transport systems described by reaction diffusion equations ». [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=983706921.

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Philipowski, Robert. « Stochastic interacting particle systems and nonlinear partial differential equations from fluid mechanics ». [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=986005622.

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Joubert, Adriaan Wolfgang. « Parallel methods for systems of nonlinear equations applied to load flow analysis ». Thesis, Queen Mary, University of London, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362721.

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Bobade, Parag Suhas. « Modeling, Approximation, and Control for a Class of Nonlinear Systems ». Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/80978.

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This work investigates modeling, approximation, estimation, and control for classes of nonlinear systems whose state evolves in space $mathbb{R}^n times H$, where $mathbb{R}^n$ is a n-dimensional Euclidean space and $H$ is a infinite dimensional Hilbert space. Specifically, two classes of nonlinear systems are studied in this dissertation. The first topic develops a novel framework for adaptive estimation of nonlinear systems using reproducing kernel Hilbert spaces. A nonlinear adaptive estimation problem is cast as a time-varying estimation problem in $mathbb{R}^d times H$. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), $H$. Furthermore, the well-posedness of the framework in the new formulation is established. We derive the sufficient conditions for existence, uniqueness, and stability of an infinite dimensional adaptive estimation problem. A condition for persistence of excitation in a RKHS in terms of an evaluation functional is introduced to establish the convergence of finite dimensional approximations of the unknown function in RKHS. Lastly, a numerical validation of this framework is presented, which could have potential applications in terrain mapping algorithms. The second topic delves into estimation and control of history dependent differential equations. This study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The governing dynamics are modeled using a specific form of functional differential equations. The class of history dependent differential equations in this work is constructed using integral operators that depend on distributed parameters. Consequently, the resulting estimation and control equations define a distributed parameter system whose state, and distributed parameters evolve in finite and infinite dimensional spaces, respectively. The well-posedness of the governing equations is established by deriving sufficient conditions for existence, uniqueness and stability for the class of functional differential equations. The error estimates for multiwavelet approximation of such history dependent operators are derived. These estimates help determine the rate of convergence of finite dimensional approximations of the online estimation equations to the infinite dimensional solution of distributed parameter system. At last, we present the adaptive sliding mode control strategy developed for the history dependent functional differential equations and numerically validate the results on a simplified pitch-plunge wing model.
Ph. D.

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24

Ali, Jaffar. « Multiple positive solutions for classes of elliptic systems with combined nonlinear effects ». Diss., Mississippi State : Mississippi State University, 2008. http://library.msstate.edu/etd/show.asp?etd=etd-07082008-153843.

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Ye, Jinglong. « Infinite semipositone systems ». Diss., Mississippi State : Mississippi State University, 2009. http://library.msstate.edu/etd/show.asp?etd=etd-07072009-132254.

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Ahlip, Rehez Ajmal. « Stability & ; filtering of stochastic systems ». Thesis, Queensland University of Technology, 1997.

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O'Day, Joseph Patrick. « Investigation of a coupled Duffing oscillator system in a varying potential field / ». Online version of thesis, 2005. https://ritdml.rit.edu/dspace/handle/1850/1212.

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Clinger, Richard A. « Stability Analysis of Systems of Difference Equations ». VCU Scholars Compass, 2007. http://hdl.handle.net/10156/1318.

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Marahrens, Daniel. « On some nonlinear partial differential equations for classical and quantum many body systems ». Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/244203.

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This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

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Taylor, Gillian Clare. « Application of the Hodgkin-Huxley equations to the propagation of small graded potentials in neurons ». Thesis, Heriot-Watt University, 1996. http://hdl.handle.net/10399/742.

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Khames, Imene. « Nonlinear network wave equations : periodic solutions and graph characterizations ». Thesis, Normandie, 2018. http://www.theses.fr/2018NORMIR04/document.

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Dans cette thèse, nous étudions les équations d’ondes non-linéaires discrètes dans des réseaux finis arbitraires. C’est un modèle général, où le Laplacien continu est remplacé par le Laplacien de graphe. Nous considérons une telle équation d’onde avec une non-linéarité cubique sur les nœuds du graphe, qui est le modèle φ4 discret, décrivant un réseau mécanique d’oscillateurs non-linéaires couplés ou un réseau électrique où les composantes sont des diodes ou des jonctions Josephson. L’équation d’onde linéaire est bien comprise en termes de modes normaux, ce sont des solutions périodiques associées aux vecteurs propres du Laplacien de graphe. Notre premier objectif est d’étudier la continuation des modes normaux dans le régime non-linéaire et le couplage des modes en présence de la non-linéarité. En inspectant les modes normaux du Laplacien de graphe, nous identifions ceux qui peuvent être étendus à des orbites périodiques non-linéaires. Il s’agit des modes normaux dont les vecteurs propres du Laplacien sont composés uniquement de {1}, {-1,+1} ou {-1,0,+1}. Nous effectuons systématiquement une analyse de stabilité linéaire (Floquet) de ces orbites et montrons le couplage des modes lorsque l’orbite est instable. Ensuite, nous caractérisons tous les graphes pour lesquels il existe des vecteurs propres du Laplacien ayant tous leurs composantes dans {-1,+1} ou {-1,0,+1}, en utilisant la théorie spectrale des graphes. Dans la deuxième partie, nous étudions des solutions périodiques localisées spatialement. En supposant une condition initiale de grande amplitude localisée sur un nœud du graphe, nous approchons l’évolution du système par l’équation de Duffing pour le nœud excité et un système linéaire forcé pour le reste du réseau. Cette approximation est validée en réduisant l’équation φ4 discrète à l’équation de Schrödinger non-linéaire de graphes et par l’analyse de Fourier de la solution numérique. Les résultats de cette thèse relient la dynamique non-linéaire à la théorie spectrale des graphes
In this thesis, we study the discrete nonlinear wave equations in arbitrary finite networks. This is a general model, where the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic on-site nonlinearity which is the discrete φ4 model, describing a mechanical network of coupled nonlinear oscillators or an electrical network where the components are diodes or Josephson junctions. The linear graph wave equation is well understood in terms of normal modes, these are periodic solutions associated to the eigenvectors of the graph Laplacian. Our first goal is to investigate the continuation of normal modes in the nonlinear regime and the modes coupling in the presence of nonlinearity. By inspecting the normal modes of the graph Laplacian, we identify which ones can be extended into nonlinear periodic orbits. They are normal modes whose Laplacian eigenvectors are composed uniquely of {1}, {-1,+1} or {-1,0,+1}. We perform a systematic linear stability (Floquet) analysis of these orbits and show the modes coupling when the orbit is unstable. Then, we characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+1}, using graph spectral theory. In the second part, we investigate periodic solutions that are spatially localized. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the discrete φ4 equation to the graph nonlinear Schrödinger equation and by Fourier analysis. The results of this thesis relate nonlinear dynamics to graph spectral theory

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Zhang, Xiaohong. « Optimal feedback control for nonlinear discrete systems and applications to optimal control of nonlinear periodic ordinary differential equations ». Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40185.

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Steinhauer, Mark. « On analysis of some nonlinear systems of partial differential equations of continuum mechanics ». Bonn : Mathematisches Institut der Universität, 2003. http://catalog.hathitrust.org/api/volumes/oclc/54890745.html.

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Chen, Yahao. « Geometric analysis of differential-algebraic equations and control systems : linear, nonlinear and linearizable ». Thesis, Normandie, 2019. http://www.theses.fr/2019NORMIR04.

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Dans la première partie de cette thèse, nous étudions les équations différentielles algébriques (en abrégé EDA) linéaires et les systèmes de contrôles linéaires associés (en abrégé SCEDA). Les problèmes traités et les résultats obtenus sont résumés comme suit : 1. Relations géométriques entre les EDA linéaires et les systèmes de contrôles génériques SCEDO. Nous introduisons une méthode, appelée explicitation, pour associer un SCEDO à n'importe quel EDA linéaire. L'explicitation d'une EDA est une classe des SCEDO, précisément un SCEDO défini, à un changement de coordonnées près, une transformation de bouclage près et une injection de sortie près. Puis nous comparons les « suites de Wong » d'une EDA avec les espaces invariants de son explicitation. Nous prouvons que la forme canonique de Kronecker FCK d'une EDA linéaire et la forme canonique de Morse FCM d'un SCEDO, ont une correspondance une à une et que leurs invariants sont liés. De plus, nous définissons l'équivalence interne de deux EDA et montrons sa particularité par rapport à l'équivalence externe en examinant les relations avec la régularité interne, i.e., l'existence et l'unicité de solutions. 2. Transformation d'un SCEDA linéaire vers sa forme canonique via la méthode d'explicitation avec des variables de driving. Nous étudions les relations entre la forme canonique par bouclage FCFB d'un SCEDA proposée dans la littérature et la forme canonique de Morse pour les SCEDO. Premièrement, dans le but de relier SCEDA avec les SCEDO, nous utilisons une méthode appelée explicitation (avec des variables de driving). Cette méthode attache à une classe de SCEDO avec deux types d'entrées (le contrôle original et le vecteur des variables de driving) à un SCEDA donné. D'autre part, pour un SCEDO linéaire classique (sans variable de driving) nous proposons une forme de Morse triangulaire FMT pour modifier la construction de la FCM. Basé sur la FMT nous proposons une forme étendue FMT et une forme étendue de FCM pour les SCEDO avec deux types d'entrées. Finalement, un algorithme est donné pour transformer un SCEDA dans sa FCFB. Cet algorithme est construit sur la FCM d'un SCEDO donné par la procédure d'explicitation. Un exemple numérique illustre la structure et l'efficacité de l'algorithme. Pour les EDA non linéaires et les SCEDA (quasi linéaires) nous étudions les problèmes suivants : 3. Explicitations, analyse externe et interne et formes normales des EDA non linéaires. Nous généralisons les deux procédures d'explicitation (avec ou sans variables de driving) dans le cas des EDA non linéaires. L'objectif de ces deux méthodes est d'associer un SCEDO non linéaire à une EDA non linéaire telle que nous puissions l'analyser à l'aide de la théorie des EDO non linéaires. Nous comparons les différences de l'équivalence interne et externe des EDA non linéaires en étudiant leurs relations avec l'existence et l'unicité d'une solution (régularité interne). Puis nous montrons que l'analyse interne des EDA non linéaire est liée à la dynamique nulle en théorie classique du contrôle non linéaire. De plus, nous montrons les relations des EDAS de forme purement semi-explicite avec les 2 procédures d'explicitations. Finalement, une généralisation de la forme de Weierstrass non linéaire FW basée sur la dynamique nulle d'un SCEDO non linéaire donné par la méthode d'explicitation est proposée
In the first part of this thesis, we study linear differential-algebraic equations (shortly, DAEs) and linear control systems given by DAEs (shortly, DAECSs). The discussed problems and obtained results are summarized as follows. 1. Geometric connections between linear DAEs and linear ODE control systems ODECSs. We propose a procedure, named explicitation, to associate a linear ODECS to any linear DAE. The explicitation of a DAE is a class of ODECSs, or more precisely, an ODECS defined up to a coordinates change, a feedback transformation and an output injection. Then we compare the Wong sequences of a DAE with invariant subspaces of its explicitation. We prove that the basic canonical forms, the Kronecker canonical form KCF of linear DAEs and the Morse canonical form MCF of ODECSs, have a perfect correspondence and their invariants (indices and subspaces) are related. Furthermore, we define the internal equivalence of two DAEs and show its difference with the external equivalence by discussing their relations with internal regularity, i.e., the existence and uniqueness of solutions. 2. Transform a linear DAECS into its feedback canonical form via the explicitation with driving variables. We study connections between the feedback canonical form FBCF of DAE control systems DAECSs proposed in the literature and the famous Morse canonical form MCF of ODECSs. In order to connect DAECSs with ODECSs, we use a procedure named explicitation (with driving variables). This procedure attaches a class of ODECSs with two kinds of inputs (the original control input and the vector of driving variables) to a given DAECS. On the other hand, for classical linear ODECSs (without driving variables), we propose a Morse triangular form MTF to modify the construction of the classical MCF. Based on the MTF, we propose an extended MTF and an extended MCF for ODECSs with two kinds of inputs. Finally, an algorithm is proposed to transform a given DAECS into its FBCF. This algorithm is based on the extended MCF of an ODECS given by the explication procedure. Finally, a numerical example is given to show the structure and efficiency of the proposed algorithm. For nonlinear DAEs and DAECSs (of quasi-linear form), we study the following problems: 3. Explicitations, external and internal analysis, and normal forms of nonlinear DAEs. We generalize the two explicitation procedures (with or without driving variable) proposed in the linear case for nonlinear DAEs of quasi-linear form. The purpose of these two explicitation procedures is to associate a nonlinear ODECS to any nonlinear DAE such that we can use the classical nonlinear ODE control theory to analyze nonlinear DAEs. We discuss differences of internal and external equivalence of nonlinear DAEs by showing their relations with the existence and uniqueness of solutions (internal regularity). Then we show that the internal analysis of nonlinear DAEs is closely related to the zero dynamics in the classical nonlinear control theory. Moreover, we show relations of DAEs of pure semi-explicit form with the two explicitation procedures. Furthermore, a nonlinear generalization of the Weierstrass form WE is proposed based on the zero dynamics of a nonlinear ODECS given by the explicitation procedure

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Zhang, Jin. « Identification of nonlinear structural dynamical system ». Diss., Georgia Institute of Technology, 1994. http://hdl.handle.net/1853/12270.

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Di, Cosmo Jonathan. « Nonlinear Schrödinger equation and Schrödinger-Poisson system in the semiclassical limit ». Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209863.

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The nonlinear Schrödinger equation appears in different fields of physics, for example in the theory of Bose-Einstein condensates or in wave propagation models. From a mathematical point of view, the study of this equation is interesting and delicate, notably because it can have a very rich set of solutions with various behaviours.

In this thesis, we have been interested in standing waves, which satisfy an elliptic partial differential equation. When this equation is seen as a singularly perturbed problem, its solutions concentrate, in the sense that they converge uniformly to zero outside some concentration set, while they remain positive on this set.

We have obtained three kind of new results. Firstly, under symmetry assumptions, we have found solutions concentrating on a sphere. Secondly, we have obtained the same type of solutions for the Schrödinger-Poisson system. The method consists in applying the mountain pass theorem to a penalized problem. Thirdly, we have proved the existence of solutions of the nonlinear Schrödinger equation concentrating at a local maximum of the potential. These solutions are found by a more general minimax principle. Our results are characterized by very weak assumptions on the potential./

L'équation de Schrödinger non-linéaire apparaît dans différents domaines de la physique, par exemple dans la théorie des condensats de Bose-Einstein ou dans des modèles de propagation d'ondes. D'un point de vue mathématique, l'étude de cette équation est intéressante et délicate, notamment parce qu'elle peut posséder un ensemble très riche de solutions avec des comportements variés.

Dans cette thèse ,nous nous sommes intéressés aux ondes stationnaires, qui satisfont une équation aux dérivées partielles elliptique. Lorsque cette équation est vue comme un problème de perturbations singulières, ses solutions se concentrent, dans le sens où elles tendent uniformément vers zéro en dehors d'un certain ensemble de concentration, tout en restant positives sur cet ensemble.

Nous avons obtenu trois types de résultats nouveaux. Premièrement, sous des hypothèses de symétrie, nous avons trouvé des solutions qui se concentrent sur une sphère. Deuxièmement, nous avons obtenu le même type de solutions pour le système de Schrödinger-Poisson. La méthode consiste à appliquer le théorème du col à un problème pénalisé. Troisièmement, nous avons démontré l'existence de solutions de l'équation de Schrödinger non-linéaire qui se concentrent en un maximum local du potentiel. Ces solutions sont obtenues par un principe de minimax plus général. Nos résultats se caractérisent par des hypothèses très faibles sur le potentiel.
Doctorat en sciences, Spécialisation mathématiques
info:eu-repo/semantics/nonPublished

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Chan, Chuen Lit. « A perturbation-incremental (PI) method for strongly non-linear oscillators and systems of delay differential equations / ». access full-text access abstract and table of contents, 2005. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?phd-ma-b1988767xa.pdf.

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Thesis (Ph.D.)--City University of Hong Kong, 2005.
"Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy" Includes bibliographical references (leaves 121-133)

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Lazaryan, Shushan, Nika LAzaryan et Nika Lazaryan. « Discrete Nonlinear Planar Systems and Applications to Biological Population Models ». VCU Scholars Compass, 2015. http://scholarscompass.vcu.edu/etd/4025.

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We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates.

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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.

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Zhao, Yaxi. « Numerical solutions of nonlinear parabolic problems using combined-block iterative methods / ». Electronic version (PDF), 2003. http://dl.uncw.edu/etd/2003/zhaoy/yaxizhao.pdf.

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Pereira, da Silva João Gustavo [Verfasser]. « Modelling Flash Sintering Using Systems of Nonlinear Differential Equations / João Gustavo Pereira da Silva ». München : Verlag Dr. Hut, 2019. http://d-nb.info/1190421836/34.

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Sakurai, Atsunori. « Exploring Nonlinear Responses of Quantum Dissipative Systems from Reduced Hierarchy Equations of Motion Approach ». 京都大学 (Kyoto University), 2013. http://hdl.handle.net/2433/179368.

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Yang, Lixiang. « Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations ». The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979.

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Gerdin, Markus. « Identification and Estimation for Models Described by Differential-Algebraic Equations ». Doctoral thesis, Linköping : Department of Electrical Engineering, Linköpings universitet, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-7600.

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Cimen, Tayfun. « Global optimal feedback control of nonlinear systems and viscosity solutions of Hamilton-Jacobi-Bellman equations ». Thesis, University of Sheffield, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.289660.

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Smith, Todd Blanton. « Variational embedded solitons, and traveling wavetrains generated by generalized Hopf bifurcations, in some NLPDE systems ». Doctoral diss., University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5042.

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In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diffusion equations with arbitrary nonlinear/reaction terms. For specific choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors.; These predictions are verified and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at infinity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
ID: 029809927; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 121-129).
Ph.D.
Doctorate
Mathematics
Sciences

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Gilmour, Isla. « Nonlinear model evaluation : ɩ-shadowing, probabilistic prediction and weather forecasting ». Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298797.

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Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of ɩ-shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in every observation, including the initial condition, ɩ-shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one. In a perfect model scenario a good variational assimilation technique will yield an ɩ-shadowing trajectory but this is not the case given an imperfect model; the inability of the model to ɩ-shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement. While the ɩ-shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model.

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SOAVE, NICOLA. « Variational and geometric methods for nonlinear differential equations ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/49889.

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This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.

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Durmaz, Burak. « Sliding Mode Control Of Linearly Actuated Nonlinear Systems ». Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12610666/index.pdf.

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This study covers the sliding mode control design for a class of nonlinear systems, where the control input affects the state of the system linearly as described by (d/dt)x=A(x)x+B(x)u+d(x). The main streamline of the study is the sliding surface design for the system. Since there is no systematic way of designing sliding surfaces for nonlinear systems, a moving sliding surface is designed such that its parameters are determined in an adaptive manner to cope with the nonlinearities of the system. This adaptive manner includes only the automatic adaptation of the sliding surface by determining its parameters by means of solving the State Dependent Riccati Equations (SDRE) online during the control process. The two methods developed in this study: SDRE combined sliding control and the pure SDRE with bias terms are applied to a longitudinal model of a generic hypersonic air vehicle to compare the results.

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Bradley, Aoibhinn Maire. « Analysis of nonlinear spatio-temporal partial differential equations : applications to host-parasite systems and bubble growth ». Thesis, University of Strathclyde, 2014. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=24405.

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The mountain hare population currently appears to be under threat in Scotland. The natural population cycles exhibited by this species are thought to be, at least in part, due to its infestation by a parasitic worm. We seek to gain an understanding of these population dynamics through a mathematical model of this system and so determine whether low population levels observed in the field are a natural trough associated with this cycling, or whether they point to a more serious decline in overall population densities. A generic result, that can be used to predict the presence of periodic travelling waves (PTWs) in a spatially heterogeneous system, is reported. This result is applicable to any two population host-parasite system with a supercritical Hopf bifurcation in the reaction kinetics. Application of this result to two examples of well studied host-parasite systems, namely the mountain hare and the red grouse systems, predicts and illustrates, for the first time, the existence of PTWs as solutions for these reaction advection diffusion schemes. One method for designing bone scaffolds involves the acoustic irradiation of a reacting polymer foam resulting in a final sample with graded porosity. The work in this thesis represents the first attempt to derive a mathematical model, for this empirical method, in order to inform the experimental design and tailor the porosity profile of samples. We isolate and study the direct effect of the acoustic pressure amplitude as well as its indirect effect on the reaction rate. We demonstrate that the direct effect of the acoustic pressure amplitude is negligible due to a high degree of attenuation by the sample. The indirect effect, on reaction rate, is significant and the standing wave is shown to produce a heterogeneous bubble size distribution. Several suggestions for further work are made.

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