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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: 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: 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: 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: Àlex Haro Publisher: Springer ISBN: 9783319296609 Category : Mathematics Languages : en Pages : 267
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: Kaspar Nipp Publisher: ISBN: 9783037191248 Category : Mathematics Languages : en Pages : 216
Book Description
In this book, dynamical systems are investigated from a geometric viewpoint. Admitting an invariant manifold is a strong geometric property of a dynamical system. This text presents rigorous results on invariant manifolds and gives examples of possible applications. In the first part, discrete dynamical systems in Banach spaces are considered. Results on the existence and smoothness of attractive and repulsive invariant manifolds are derived. In addition, perturbations and approximations of the manifolds and the foliation of the adjacent space are treated. In the second part, analogous results for continuous dynamical systems in finite dimensions are established. In the third part, the theory developed is applied to problems in numerical analysis and to singularly perturbed systems of ordinary differential equations. The mathematical approach is based on the so-called graph transform, already used by Hadamard in 1901. The aim is to establish invariant manifold results in a simple setting that provides quantitative estimates. The book is targeted at researchers in the field of dynamical systems interested in precise theorems that are easy to apply. The application part might also serve as an underlying text for a student seminar in mathematics.
Author: I. U. Bronshte?n Publisher: World Scientific ISBN: 9789810215729 Category : Mathematics Languages : en Pages : 408
Book Description
This book deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.
Author: Rui Chen
Author: Rui Chen
Author: Zhikai Wang
Author: Àlex Haro
Author: Kenji Nakanishi
Author: Àlex Haro
Author: Kaspar Nipp
Author: I. U. Bronshte?n
Author: University of Minnesota. Institute for Mathematics and Its Applications
Author: M. W. Hirsch
Author: Ronald Dale Haynes