Dissertations / Theses: 'Nonlinear Dynamic Equations'

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Peters, James Edward II. "Group analysis of the nonlinear dynamic equations of elastic strings." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/29348.

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Sotoudeh, Zahra. "Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41179.

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Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis. This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables. In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research. This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.

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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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Zigic, Jovan. "Optimization Methods for Dynamic Mode Decomposition of Nonlinear Partial Differential Equations." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103862.

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Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.
Master of Science
The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

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Brown, Andrew M. "Design, construction and analysis of a chaotic vibratory system." Thesis, Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/18172.

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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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Qu, Zheng. "Nonlinear Perron-Frobenius theory and max-plus numerical methods for Hamilton-Jacobi equations." Palaiseau, Ecole polytechnique, 2013. http://pastel.archives-ouvertes.fr/docs/00/92/71/22/PDF/thesis.pdf.

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Une approche fondamentale pour la résolution de problémes de contrôle optimal est basée sur le principe de programmation dynamique. Ce principe conduit aux équations d'Hamilton-Jacobi, qui peuvent être résolues numériquement par des méthodes classiques comme la méthode des différences finies, les méthodes semi-lagrangiennes, ou les schémas antidiffusifs. À cause de la discrétisation de l'espace d'état, la dimension des problèmes de contrôle pouvant être abordés par ces méthodes classiques est souvent limitée à 3 ou 4. Ce phénomène est appellé malédiction de la dimension. Cette thèse porte sur les méthodes numériques max-plus en contôle optimal deterministe et ses analyses de convergence. Nous étudions et developpons des méthodes numériques destinées à attenuer la malédiction de la dimension, pour lesquelles nous obtenons des estimations théoriques de complexité. Les preuves reposent sur des résultats de théorie de Perron-Frobenius non linéaire. En particulier, nous étudions les propriétés de contraction des opérateurs monotones et non expansifs, pour différentes métriques de Finsler sur un cône (métrique de Thompson, métrique projective d'Hilbert). Nous donnons par ailleurs une généralisation du "coefficient d'ergodicité de Dobrushin" à des opérateurs de Markov sur un cône général. Nous appliquons ces résultats aux systèmes de consensus ainsi qu'aux équations de Riccati généralisées apparaissant en contrôle stochastique
Dynamic programming is one of the main approaches to solve optimal control problems. It reduces the latter problems to Hamilton-Jacobi partial differential equations (PDE). Several techniques have been proposed in the literature to solve these PDE. We mention, for example, finite difference schemes, the so-called discrete dynamic programming method or semi-Lagrangian method, or the antidiffusive schemes. All these methods are grid-based, i. E. , they require a discretization of the state space, and thus suffer from the so-called curse of dimensionality. The present thesis focuses on max-plus numerical solutions and convergence analysis for medium to high dimensional deterministic optimal control problems. We develop here max-plus based numerical algorithms for which we establish theoretical complexity estimates. The proof of these estimates is based on results of nonlinear Perron-Frobenius theory. In particular, we study the contraction properties of monotone or non-expansive nonlinear operators, with respect to several classical metrics on cones (Thompson's metric, Hilbert's projective metric), and obtain nonlinear or non-commutative generalizations of the "ergodicity coefficients" arising in the theory of Markov chains. These results have applications in consensus theory and also to the generalized Riccati equations arising in stochastic optimal control

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Ferrara, Joseph. "A Study of Nonlinear Dynamics in Mathematical Biology." UNF Digital Commons, 2013. http://digitalcommons.unf.edu/etd/448.

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We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

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Larson, David F. H. "Modeling nonlinear stochastic ocean loads as diffusive stochastic differential equations to derive the dynamic responses of offshore wind turbines." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105690.

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Thesis: S.B., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (page 54).
A procedure is developed for modeling stochastic ocean wave and wind loads as diffusive stochastic differential equations (SDE) in a state space form to derive the response statistics of offshore structures, specifically wind turbines. Often, severe wind and wave systems are highly nonlinear and thus treatment as linear systems is not applicable, leading to computationally expensive Monte Carlo simulations. Using Stratonovich-form diffusive stochastic differential equations, both linear and nonlinear components of the wind thrust can be modeled as 2 state SDE. These processes can be superposed with both the linear and nonlinear (inertial and viscous) wave forces, also modeled as a multi-dimensional state space SDE. Furthermore, upon implementing the ESPRIT algorithm to fit the autocorrelation function of any real sea state spectrum, a simple 2-state space model can be derived to completely describe the wave forces. The resulting compound state-space SDE model forms the input to a multi-dimension state-space Fokker-Planck equation, governing the dynamical response of the wind turbine structure. Its solution yields response, fatigue and failure statistics-information critical to the design of any offshore structure. The resulting Fokker-Planck equation can be solved using existing numerical schemes.
by David F.H. Larson.
S.B.

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Challa, Subhash. "Nonlinear state estimation and filtering with applications to target tracking problems." Thesis, Queensland University of Technology, 1998.

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Pittayakanchit, Weerapat. "The Global Stability of the Solution to the Morse Potential in a Catastrophic Regime." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/hmc_theses/72.

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Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single component. We are still examining the local minimizers with multiple components to determine whether the candidate solution is the minimum-energy configuration.

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Yip, Lai-pan. "Nonlinear and localized modes in hydrodynamics and vortex dynamics." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/hkuto/record/B39316919.

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Yip, Lai-pan, and 葉禮彬. "Nonlinear and localized modes in hydrodynamics and vortex dynamics." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B39316919.

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Deshmukh, Rohit. "Model Order Reduction of Incompressible Turbulent Flows." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1471618549.

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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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Tsang, Cheng-hou Alan, and 曾正豪. "Dynamics of waves and patterns of the complex Ginburg Landau and soliton management models: localized gain andeffects of inhomogeneity." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2011. http://hub.hku.hk/bib/B46975433.

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Kwek, Keng-Huat. "On Cahn-Hilliard type equation." Diss., Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28819.

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Hays, Joseph T. "Parametric Optimal Design Of Uncertain Dynamical Systems." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/28850.

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This research effort develops a comprehensive computational framework to support the parametric optimal design of uncertain dynamical systems. Uncertainty comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; not accounting for uncertainty may result in poor robustness, sub-optimal performance and higher manufacturing costs. Contemporary methods for the quantification of uncertainty in dynamical systems are computationally intensive which, so far, have made a robust design optimization methodology prohibitive. Some existing algorithms address uncertainty in sensors and actuators during an optimal design; however, a comprehensive design framework that can treat all kinds of uncertainty with diverse distribution characteristics in a unified way is currently unavailable. The computational framework uses Generalized Polynomial Chaos methodology to quantify the effects of various sources of uncertainty found in dynamical systems; a Least-Squares Collocation Method is used to solve the corresponding uncertain differential equations. This technique is significantly faster computationally than traditional sampling methods and makes the construction of a parametric optimal design framework for uncertain systems feasible. The novel framework allows to directly treat uncertainty in the parametric optimal design process. Specifically, the following design problems are addressed: motion planning of fully-actuated and under-actuated systems; multi-objective robust design optimization; and optimal uncertainty apportionment concurrently with robust design optimization. The framework advances the state-of-the-art and enables engineers to produce more robust and optimally performing designs at an optimal manufacturing cost.
Ph. D.

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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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Lazaryan, Shushan, Nika LAzaryan, and 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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Campbell, Fiona Mary. "The dynamics of soliton interaction." Thesis, University of Strathclyde, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.248325.

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Marrekchi, Hamadi. "Dynamic compensators for a nonlinear conservation law." Diss., This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-05042006-164530/.

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Lam, Chun-kit, and 林晉傑. "The dynamics of wave propagation in an inhomogeneous medium: the complex Ginzburg-Landau model." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40887881.

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Lam, Chun-kit. "The dynamics of wave propagation in an inhomogeneous medium the complex Ginzburg-Landau model /." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40887881.

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Osman, Frederick. "Nonlinear paraxial equation at laser plasma interaction /." [Campbelltown, N.S.W. : The author], 1998. http://library.uws.edu.au/adt-NUWS/public/adt-NUWS20030707.114012/index.html.

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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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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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Qiao, Zhonghua. "Numerical solution for nonlinear Poisson-Boltzmann equations and numerical simulations for spike dynamics." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/727.

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Honda, Ethan Philip. "Resonant dynamics within the nonlinear Klein-Gordon equation : Much ado about oscillons /." Full text (PDF) from UMI/Dissertation Abstracts International, 2000. http://wwwlib.umi.com/cr/utexas/fullcit?p9992817.

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Farnum, Edward D. "Stability and dynamics of solitary waves in nonlinear optical materials /." Thesis, Connect to this title online; UW restricted, 2005. http://hdl.handle.net/1773/6766.

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Baturin, Nickolay G. "Dynamics and effects of the tropical instability waves /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1997. http://wwwlib.umi.com/cr/ucsd/fullcit?p9737308.

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Lindgren, Joseph B. "Orbital Stability Results for Soliton Solutions to Nonlinear Schrödinger Equations with External Potentials." UKnowledge, 2017. http://uknowledge.uky.edu/math_etds/46.

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For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method. For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.

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Meissen, Emily Philomena, and Emily Philomena Meissen. "Invading a Structured Population: A Bifurcation Approach." Diss., The University of Arizona, 2017. http://hdl.handle.net/10150/625610.

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Matrix population models are discrete in both time and state-space, where a matrix with density-dependent entries is used to project a population vector of a stage-structured population from one time to the next. Such models are useful for modeling populations with discrete categorizations (e.g. developmental cycles, communities of multiple species, differing sizes, etc.). We present a general matrix model of two interacting populations where one (the resident) has a stable cycle, and we analyze when the other population (the invader) can successfully invade. Specifically, we study the local bifurcations of coexistence cycles as the resident cycle destabilizes, where a cycle of length 1 corresponds to an equilibrium. We make no assumptions on the types of interactions between the populations or on the population structure of the resident; we consider when the invader's projection matrix is primitive or imprimitive and 2x2. The simplest biological scenarios for such structures are an iteroparous invader and a two-stage semelparous invader. When the invader has a primitive projection matrix, coexistence cycles (of the same period as the resident cycle) bifurcate from the resident-cycle. When the invader has an imprimitive two-stage projection matrix, two types of coexistence cycles bifurcate from the resident-cycle: cycles of the same period and cycles of double the period. In both the primitive and imprimitive cases, we provide diagnostic quantities to determine the direction of bifurcation and the stability of the bifurcating cycles. Because we only perform a local stability analysis, the only successful invasion provided by our results is through stable coexistence cycles. As we show in some simple examples, however, the invader may persist when the coexistence cycles are unstable through competitive exclusion where the branch of bifurcating cycles connects to a branch of invader attractors and creates a multi-attractor scenario known as a strong Allee effect.

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Osman, Frederick. "Nonlinear paraxial equation at laser plasma interaction." Thesis, [Campbelltown, N.S.W. : The author], 1998. http://handle.uws.edu.au:8081/1959.7/280.

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This thesis presents an investigation into the behaviour of a laser beam of finite diameter in a plasma with respect to forces and optical properties, which lead to self-focusing of the beam. The transient setting of ponderomotive nonlinearity in a collisionless plasma has been studied, and consequently the self- focusing of the pulse, and the focusing of the plasma wave occurs. The description of a self-focusing mechanism of laser radiation in the plasma due to nonlinear forces acting on the plasma in the lateral direction, relative to the laser has been investigated in the non-relativistic regime. The behaviour of the laser beams in plasma, which is the domain of self-focusing at high or moderate intensity, is dominated by the nonlinear force. The investigation of self-focusing processes of laser beams in plasma result from the relativistic mass and energy dependency of the refractive index at high laser intensities. Here the relativistic effects are considered to evaluate the relativistic self-focusing lenghts for the neodymium glass radiation, at different plasma densities of various laser intensities. A sequence of code in C++ has been developed to explore in depth self-focusing over a wide range of parameters. The nonlinear plasma dielectric function to relativistic electron motion will be derived in the latter part of this thesis. From that, one can obtain the nonlinear refractive index of the plasma and estimate the importance of relativistic self-focusing as compared to ponderomotive non-relativistic self-focusing, at very high laser intensities. When the laser intensity is very high, pondermotive self-focusing will be dominant. But at some point, when the oscillating velocity of the plasma electron becomes very large, relativistic effects will also play a role in self-focusing. A numerical and theoretical study of the generation and propagation of oscillation in the semiclassical limit of the nonlinear paraxial equation is presented in this thesis. In a general setting of both dimension and nonlinearity, the essential differences between the 'defocusing' and 'focusing' cases hence is identified. Presented in this thesis are the nonlinearity and dispersion effects involved in the propagation of solitions which can be understood by using a numerical routines were implemented through the use of the mathematica program, and results give a very clear idea of this interesting phenomena

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Jendrej, Jacek. "On the dynamics of energy-critical focusing wave equations." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX029/document.

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Cette thèse est consacrée à l’étude du comportement global des solutions de l’équation des ondes énergie-critique. On s’intéresse tout spécialement à la description de la dynamique du système dans l’espace de l’énergie. Nous développons une variante de la méthode d’énergie qui permet de construire des solutions explosives de type II, instables. Ensuite, par une démarche similaire, nous donnons le premier exemple d’une solution radiale de l’équation des ondes énergie-critique qui converge dans l’espace de l’énergie vers une superposition de deux états stationnaires (bulles). En appliquant notre méthode au cas de l’équation des ondes des applications harmoniques (wave map), nous obtenons des solutions de type bulle-antibulle, en toute classe d’équivariance k > 2. Pour l’équation des ondes énergie-critique radiale, nous étudions également le lien entre la vitesse de l’explosion de type II et la limite faible de la solution au moment de l’explosion. Finalement, nous montrons qu’il est impossible qu’une solution radiale converge vers une superposition de deux bulles ayant les signes opposés
In this thesis we study the global behavior of solutions of the energy-criticalfocusing nonlinear wave equation, with a special emphasis on the description of the dynamics in the energy space. We develop a new approach, based on the energy method, to constructing unstable type II blow-up solutions. Next, we give the first example of a radial two-bubble solution of the energy-critical wave equation. By implementing this construction in the case of the equivariant wave map equation, we obtain bubble-antibubble solutions in equivariance classes k > 2. We also study the relationship between the speed of a type II blow-up and the weak limit of the solution at the blow-up time. Finally, we prove that there are no pure radial two-bubbles with opposite signs for the energy-critical wave equation

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Kim, Tae Eun. "Quasi-solution Approach to Nonlinear Integro-differential Equations: Applications to 2-D Vortex Patch Problems." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499793039477532.

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Colinet, Pierre. "Amplitude equations and nonlinear dynamics of surface-tension and buoyancy-driven convective instabilities." Doctoral thesis, Universite Libre de Bruxelles, 1997. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212204.

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This work is a theoretical contribution to the study of thermo-hydrodynamic instabilities in fluids submitted to surface-tension (Marangoni) and buoyancy (Rayleigh) effects in layered (Benard) configurations. The driving constraint consists in a thermal (or a concentrational) gradient orthogonal to the plane of the layer(s).

Linear, weakly nonlinear as well as strongly nonlinear analyses are carried out, with emphasis on high Prandtl (or Schmidt) number fluids, although some results are also given for low-Prandtl number liquid metals. Attention is mostly devoted to the mechanisms responsible for the onset of complex spatio-temporal behaviours in these systems, as well as to the theoretical explanation of some existing experimental results.

As far as linear stability analyses (of the diffusive reference state) are concerned, a number of different effects are studied, such as Benard convection in two layers coupled at an interface (for which a general classification of instability modes is proposed), surface deformation effects and phase-change effects (non-equilibrium evaporation). Moreover, a number of different monotonous and oscillatory instability modes (leading respectively to patterns and waves in the nonlinear regime) are identified. In the case of oscillatory modes in a liquid layer with deformable interface heated from above, our analysis generalises and clarifies earlier works on the subject. A new Rayleigh-Marangoni oscillatory mode is also described for a liquid layer with an undeformable interface heated from above (coupling between internal and surface waves).

Weakly nonlinear analyses are then presented, first for monotonous modes in a 3D system. Emphasis is placed on the derivation of amplitude (Ginzburg-Landau) equations, with universal structure determined by the general symmetry properties of the physical system considered. These equations are thus valid outside the context of hydrodynamic instabilities, although they generally depend on a certain number of numerical coefficients which are calculated for the specific convective systems studied. The nonlinear competitions of patterns such as convective rolls, hexagons and squares is studied, showing the preference for hexagons with upflow at the centre in the surface-tension-driven case (and moderate Prandtl number), and of rolls in the buoyancy-induced case.

A transition to square patterns recently observed in experiments is also explained by amplitude equation analysis. The role of several fluid properties and of heat transfer conditions at the free interface is examined, for one-layer and two-layer systems. We also analyse modulation effects (spatial variation of the envelope of the patterns) in hexagonal patterns, leading to the description of secondary instabilities of supercritical hexagons (Busse balloon) in terms of phase diffusion equations, and of pentagon-heptagon defects in the hexagonal structures. In the frame of a general non-variational system of amplitude equations, we show that the pentagon-heptagon defects are generally not motionless, and may even lead to complex spatio-temporal dynamics (via a process of multiplication of defects in hexagonal structures).

The onset of waves is also studied in weakly nonlinear 2D situations. The competition between travelling and standing waves is first analysed in a two-layer Rayleigh-Benard system (competition between thermal and mechanical coupling of the layers), in the vicinity of special values of the parameters for which a multiple (Takens-Bogdanov) bifurcation occurs. The behaviours in the vicinity of this point are numerically explored. Then, the interaction between waves and steady patterns with different wavenumbers is analysed. Spatially quasiperiodic (mixed) states are found to be stable in some range when the interaction between waves and patterns is non-resonant, while several transitions to chaotic dynamics (among which an infinite sequence of homoclinic bifurcations) occur when it is resonant. Some of these results have quite general validity, because they are shown to be entirely determined by quadratic interactions in amplitude equations.

Finally, models of strongly nonlinear surface-tension-driven convection are derived and analysed, which are thought to be representative of the transitions to thermal turbulence occurring at very high driving gradient. The role of the fastest growing modes (intrinsic length scale) is discussed, as well as scalings of steady regimes and their secondary instabilities (due to instability of the thermal boundary layer), leading to chaotic spatio-temporal dynamics whose preliminary analysis (energy spectrum) reveals features characteristic of hydrodynamic turbulence. Some of the (2D and 3D) results presented are in qualitative agreement with experiments (interfacial turbulence).

Doctorat en sciences appliquées
info:eu-repo/semantics/nonPublished

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Soneson, Joshua Eric. "Optical Pulse Dynamics in Nonlinear and Resonant Nanocomposite Media." Diss., Tucson, Arizona : University of Arizona, 2005. http://etd.library.arizona.edu/etd/GetFileServlet?file=file:///data1/pdf/etd/azu%5Fetd%5F1274%5F1%5Fm.pdf&type=application/pdf.

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39

Cintra, Daniel. "Contribution à l'étude du phénomène d'oscillation argumentaire." Thesis, Paris Est, 2017. http://www.theses.fr/2017PESC1220/document.

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Contribution à l’étude du phénomène d’oscillation argumentaire. L’oscillateur argumentaire a un mouvement stable périodique, à une fréquence proche de sa fréquence fondamentale, lorsqu’il est soumis à une excitation provenant d’une source de type harmonique, à une fréquence qui est un multiple de ladite fréquence fondamentale, et agissant de manière telle que son interaction avec le système dépende des coordonnées d’espace du système. La présente thèse étudie quelques systèmes argumentaires et essaie de mettre en évidence des relations symboliques entre les paramètres de ces systèmes et leur comportement observé ou calculé. C’est la représentation de Van der Pol qui a été utilisée la plupart du temps pour représenter l’état du système, car elle est bien adaptée à la méthode de centrage, où l'on cherche une solution sous forme d’un signal de type sinusoïdal, d’amplitude et de phase lentement variables. L’originalité de la présente thèse vis-à-vis des publications antérieures est dans la modélisation, plus proche des systèmes physiques réels, dans les développements symboliques qui donnent des représentations inédites, dans le mode de réalisation des expériences, qui utilisent toutes une visualisation de Van der Pol en temps réel, et dans l’objet de l’expérience de la poutre excitée axialement de manière argumentaire. Au cours de cette thèse, des systèmes simples à un DDL ont été modélisés, construits et expérimentés. Des relations symboliques, notamment concernant les probabilités de capture par des attracteurs, ainsi que des critères de stabilité et une solution symbolique approchée, ont été mis en évidence. Un système continu constitué d’une poutre élancée excitée axialement a ensuite été modélisé à l’aide de deux modèles et expérimenté ; toujours dans le domaine symbolique, des propriétés ont été étudiées, notamment concernant des combinaisons de plages de paramètres permettant au phénomène argumentaire d’exister
Contribution to the study of the argumental oscillation phenomenon. The argumental oscillator has a stable periodic motion at a frequency close to its fundamental frequency when it is subjected to an excitation from a harmonic source at a frequency which is a multiple of said fundamental frequency, and acting in such a way that its interaction with the system depends on the space coordinates of the system. This thesis studies some argumental systems and tries to demonstrate symbolic relations between the parameters of these systems and their observed or calculated behavior. The Van der Pol representation was used most of the time to represent the state of the system, as it is well adapted to the averaging method, where a solution is sought as a signal of sinusoidal type, with slowly varying amplitude and phase. The originality of this thesis with respect to previous publications is in the modeling, closer to real physical systems, in the symbolic developments that give new representations, in the embodiment of the experiments, which all use a real-time Van der Pol visualization, and in the object of the experiment of the beam axially excited in an argumental way. During this thesis, simple systems with one DDL have been modeled, built and tested. Symbolic relationships have been highlighted, in particular with regard to the probabilities of capture by attractors, as well as stability criteria and an approximate symbolic solution. A continuous system consisting of an axially excited slender beam was then modeled using two models, and tested; still in the symbolic domain, properties have been studied, especially concerning combinations of parameter ranges allowing the argumental phenomenon to occur

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Li, Rui. "Dynamical Properties of Solutions for Various Types of Nonlinear Partial Differential Equations." Thesis, Curtin University, 2019. http://hdl.handle.net/20.500.11937/79916.

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In this thesis, we investigate the existence of global weak solutions for a generalized Benjamin-Bona-Burgers equation and a nonlinear equation with quartic nonlinearities. The existence of local weak solutions and well-posedness of local strong solutions are established for two nonlinear equations with quadratic and cubic nonlinearities, respectively. Moreover, conditions of wave breaking for a generalized Degasperis-Procesi equation are obtained.

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Palmer, Robert. "Dynamics and Asymptotic Behavior of the Solutions of a Nonlinear Differential Equation." TopSCHOLAR®, 1999. http://digitalcommons.wku.edu/theses/737.

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Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to determine its uniqueness, range of existence, and dependency upon initial conditions. The long - term behavior of the solution is also determined. Solutions of the initial value problem are also analyzed when p is an integer greater than 2. Conclusions about the behavior of solutions to such initial value problems are made, and such conclusions depend in part upon whether p is even or odd. Mathematica Version 3.0 was unable to determine formulas for selected problems of this form.

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Inui, Takahisa. "GLOBAL DYNAMICS OF SOLUTIONS WITH GROUP INVARIANCE FOR THE NONLINEAR SCHRODINGER EQUATION." 京都大学 (Kyoto University), 2017. http://hdl.handle.net/2433/225377.

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Yang, Kai. "Dynamics of the energy critical nonlinear Schrödinger equation with inverse square potential." Diss., University of Iowa, 2017. https://ir.uiowa.edu/etd/5685.

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We consider the Cauchy problem for the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of the kinetic energy and potential energy, is conserved for all time. Due to the focusing nature, solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or even have mixed behaviors. Our goal in this thesis is to fully characterize the solution when the energy is below or at the level of the energy of the ground state solution $W_a$. Our main result contains two parts. First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters. Second, we show a rigidity result at the level of ground state solution. We prove that among all solutions with the same energy as the ground state solution, there are only two (up to symmetries) solutions $W_a^+, W_a^-$ that are exponential close to $W_a$ and serve as the threshold of scattering and blow-up. All solutions with the same energy will blow up both forward and backward in time if they go beyond the upper threshold $W_a^+$; all solutions with the same energy will scatter both forward and backward in time if they fall below the lower threshold $W_a^-$. In the case of NLS with no potential, this type of results was first obtained by Kenig-Merle \cite{R: Kenig focusing} and Duyckaerts-Merle \cite{R: D Merle}. However, as the potential has the same scaling as $\Delta$, one can not expect to extend their results in a simple perturbative way. We develop crucial spectral estimates for the operator $-\Delta+a/|x|^2$, we also rely heavily on the recent understanding of the operator $-\Delta+a/|x|^2$ in \cite{R: Harmonic inverse KMVZZ}.

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Grice, Glenn Noel Mathematics UNSW. "Constant speed flows and the nonlinear Schr??dinger equation." Awarded by:University of New South Wales. Mathematics, 2004. http://handle.unsw.edu.au/1959.4/20509.

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This thesis demonstrates how the geometric connection between the integrable Heisenberg spin equation, the nonlinear Schr??dinger equation and fluid flows with constant velocity magnitude along individual streamlines may be exploited. Specifically, we are able to construct explicitly the complete class of constant speed flows where the constant pressure surfaces constitute surfaces of revolution. This class is undoubtedly important as it contains many of the specific cases discussed earlier by other authors.

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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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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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Madurasinghe, M. A. D. "Splashless ship bows and waveless sterns /." Title page, summary, contents and general introduction only, 1986. http://web4.library.adelaide.edu.au/theses/09PH/09phm183.pdf.

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ORTOLEVA, CECILIA MARIA. "Asymptotic properties of the dynamics near stationary solutions for some nonlinear schro dinger equations." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2013. http://hdl.handle.net/10281/41846.

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The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The rst model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a point (or contact) interaction with strength , which consists of a singular perturbation of the Laplacian described by a selfadjoint operator H , and letting the strength depend on the wave function: i du dt = H u, = (u). It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to jx x0j1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e. the coe cient of its singular part, then, in order to introduce a nonlinearity, we let the strength depend on u according to the law = jqj , with > 0. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form u(t) = ei!t !, which are orbitally stable in the range 2 (0; 1), and orbitally unstable for 1: Moreover, we show that for 2 (0; p1 2 ) [ p1 2 ; p 3+1 2 p 2 every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted Lp space which allows dispersive estimates, the following resolution holds: u(t) = ei!1t+il(t) !1 + Ut 1 + r1, where Ut is the free Schrödinger propagator, !1 > 0 and 1, r1 2 L2(R3) with kr1kL2 = O(tp) as t ! +1, p = 5 4 , 1 4 depending on 2 (0; 1= p 2), 2 (1= p 2; 1), respectively, and nally l(t) is a logarithmic increasing function that appears when 2 (p1 2 ; ), for a certain 2 p1 2 ; p 3+1 2 p 2 i . Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation i du dt = u juj4u. In this case we prove, for any and 0 su ciently small, the existence of radial nite energy solutions of the form u(t; x) = ei (t) 1=2(t)W( (t)x) + ei t + o _H1(1) as t ! +1, where (t) = 0 ln t, (t) = t , W(x) = (1+ 1 3 jxj2)1=2 is the ground state and is arbitrarily small in _H 1.

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Abell, K. A. "Analysis and computation of the dynamics of spatially discrete phase transition equations." Thesis, University of Sussex, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.344064.

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Dadashi, Shirin. "Modeling and Approximation of Nonlinear Dynamics of Flapping Flight." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78224.

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The first and most imperative step when designing a biologically inspired robot is to identify the underlying mechanics of the system or animal of interest. It is most common, perhaps, that this process generates a set of coupled nonlinear ordinary or partial differential equations. For this class of systems, the models derived from morphology of the skeleton are usually very high dimensional, nonlinear, and complex. This is particularly true if joint and link flexibility are included in the model. In addition to complexities that arise from morphology of the animal, some of the external forces that influence the dynamics of animal motion are very hard to model. A very well-established example of these forces is the unsteady aerodynamic forces applied to the wings and the body of insects, birds, and bats. These forces result from the interaction of the flapping motion of the wing and the surround- ing air. These forces generate lift and drag during flapping flight regime. As a result, they play a significant role in the description of the physics that underlies such systems. In this research we focus on dynamic and kinematic models that govern the motion of ground based robots that emulate flapping flight. The restriction to ground based biologically inspired robotic systems is predicated on two observations. First, it has become increasingly popular to design and fabricate bio-inspired robots for wind tunnel studies. Second, by restricting the robotic systems to be anchored in an inertial frame, the robotic equations of motion are well understood, and we can focus attention on flapping wing aerodynamics for such nonlinear systems. We study nonlinear modeling, identification, and control problems that feature the above complexities. This document summarizes research progress and plans that focuses on two key aspects of modeling, identification, and control of nonlinear dynamics associated with flapping flight.
Ph. D.

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Dissertations / Theses on the topic 'Nonlinear Dynamic Equations'

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