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Author: Zhikai Wang Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
In this thesis, we start with the basic concepts of dynamical systems. Then we introduce the general types of problems that the well-known software package AUTO solves. AUTO uses a boundary value algorithm with Gauss collocation and pseudo-arclength continuation. These two features distinguish AUTO from other general ODE solvers for dynamical systems. In order to compute 2D solution manifolds, AUTO uses orbit continuation. With these tools, we study two famous problems, the Lorenz system and the Circular Restricted Three-Body Problem (CR3BP). We briefly discuss the basic bifurcation and stability analysis of general ODE systems. The numerical analysis of the two problems leads to the newest algorithm to compute the 2D stable manifold of the origin of the Lorenz system and the 2D unstable manifold of appropriate periodic orbits of the CR3BP. We utilize Python for the flow control of AUTO. We also implement two visualization packages, QTPlaut and MATPlaut. They make possible the processing of large quantities of AUTO solution data with the OpenGL graphical library, dynamic memory allocation and interpolation methods. We conclude with prospect for future research.
Author: Zhikai Wang Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
In this thesis, we start with the basic concepts of dynamical systems. Then we introduce the general types of problems that the well-known software package AUTO solves. AUTO uses a boundary value algorithm with Gauss collocation and pseudo-arclength continuation. These two features distinguish AUTO from other general ODE solvers for dynamical systems. In order to compute 2D solution manifolds, AUTO uses orbit continuation. With these tools, we study two famous problems, the Lorenz system and the Circular Restricted Three-Body Problem (CR3BP). We briefly discuss the basic bifurcation and stability analysis of general ODE systems. The numerical analysis of the two problems leads to the newest algorithm to compute the 2D stable manifold of the origin of the Lorenz system and the 2D unstable manifold of appropriate periodic orbits of the CR3BP. We utilize Python for the flow control of AUTO. We also implement two visualization packages, QTPlaut and MATPlaut. They make possible the processing of large quantities of AUTO solution data with the OpenGL graphical library, dynamic memory allocation and interpolation methods. We conclude with prospect for future research.
Author: Àlex Haro Publisher: Springer ISBN: 3319296620 Category : Mathematics Languages : en Pages : 280
Book Description
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
Author: Rui Chen Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
There is much interest recently in the computation of invariant manifolds of vector fields. This thesis presents a continuation method for computing two-dimensional stable and unstable manifolds arising in dynamical systems. The computations are illustrated using a well-known example, namely, the Lorenz system, for which detailed results are presented for the so-called "Lorenz manifold", i.e., the two-dimensional stable manifold of the origin, as well as for the two-dimensional unstable manifold of one of the two symmetric nonzero equilibria. A number of the infinitely many intersection curves of these manifolds are also determined accurately. All computations are carried out using the numerical continuation software AUTO, specifically its most recent version, AUTO-07p. Various diagrams are given to illustrate the numerical results. Software based on OpenGL and Glut has been developed to visualize the numerically computed manifolds, using a triangulation of the data computed with AUTO.
Author: Hans-Christian Hege Publisher: Springer Science & Business Media ISBN: 3540886060 Category : Mathematics Languages : en Pages : 194
Book Description
Visualization research aims to provide insight into large, complicated data sets and the phenomena behind them. While there are di?erent methods of reaching this goal, topological methods stand out for their solid mathem- ical foundation, which guides the algorithmic analysis and its presentation. Topology-based methods in visualization have been around since the beg- ning of visualization as a scienti?c discipline, but they initially played only a minor role. In recent years,interest in topology-basedvisualization has grown andsigni?cantinnovationhasledto newconceptsandsuccessfulapplications. The latest trends adapt basic topological concepts to precisely express user interests in topological properties of the data. This book is the outcome of the second workshop on Topological Methods in Visualization, which was held March 4–6, 2007 in Kloster Nimbschen near Leipzig,Germany.Theworkshopbroughttogethermorethan40international researchers to present and discuss the state of the art and new trends in the ?eld of topology-based visualization. Two inspiring invited talks by George Haller, MIT, and Nelson Max, LLNL, were accompanied by 14 presentations by participants and two panel discussions on current and future trends in visualization research. This book contains thirteen research papers that have been peer-reviewed in a two-stage review process. In the ?rst phase, submitted papers where peer-reviewed by the international program committee. After the workshop accepted papers went through a revision and a second review process taking into account comments from the ?rst round and discussions at the workshop. Abouthalfthepapersconcerntopology-basedanalysisandvisualizationof ?uid?owsimulations;twopapersconcernmoregeneraltopologicalalgorithms, while the remaining papers discuss topology-based visualization methods in application areas like biology, medical imaging and electromagnetism.
Author: Peter W. Bates Publisher: American Mathematical Soc. ISBN: 0821808680 Category : Mathematics Languages : en Pages : 145
Book Description
Extends the theory for normally hyperbolic invariant manifolds to infinite dimensional dynamical systems in a Banach space, thereby providing tools for the study of PDE's and other infinite dimensional equations of evolution. In the process, the authors establish the existence of center-unstable and center-stable manifolds in a neighborhood of the unperturbed compact manifold. No index. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Kenji Nakanishi Publisher: European Mathematical Society ISBN: 9783037190951 Category : Hamiltonian systems Languages : en Pages : 264
Book Description
The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein-Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. The authors' entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors. The proofs rely on an interplay between the variational structure of the ground states and the nonlinear hyperbolic dynamics near these states. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein-Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
Author: Henk Broer Publisher: Springer ISBN: 354036398X Category : Mathematics Languages : en Pages : 178
Book Description
The authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.
Author: Stephen Wiggins Publisher: Springer Science & Business Media ISBN: 1461243122 Category : Mathematics Languages : en Pages : 198
Book Description
In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally hyperbolic invariant manifolds and foliations of normally hyperbolic invariant manifolds. In recent years these techniques have been used for the development of global perturbation methods, the study of resonance phenomena in coupled oscillators, geometric singular perturbation theory, and the study of bursting phenomena in biological oscillators. "Invariant manifold theorems" have become standard tools for applied mathematicians, physicists, engineers, and virtually anyone working on nonlinear problems from a geometric viewpoint. In this book, the author gives a self-contained development of these ideas as well as proofs of the main theorems along the lines of the seminal works of Fenichel. In general, the Fenichel theory is very valuable for many applications, but it is not easy for people to get into from existing literature. This book provides an excellent avenue to that. Wiggins also describes a variety of settings where these techniques can be used in applications.
Author: Zhikai Wang
Author: Zhikai Wang
Author: Àlex Haro
Author: Rui Chen
Author: Hans-Christian Hege
Author: Peter W. Bates
Author: Kenji Nakanishi
Author: Henk Broer
Author: Stephen Wiggins
Author: M.W. Hirsch
Author: