Dynamical Systems And Linear Algebra

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This book provides an introduction to the interplay between linear algebra and dynamical systems in continuous time and in discrete time. It first reviews the autonomous case for one matrix A via induced dynamical systems in ℝd and on Grassmannian manifolds. Then the main nonautonomous approaches are presented for which the time dependency of A(t) is given via skew-product flows using periodicity, or topological (chain recurrence) or ergodic properties (invariant measures). The authors develop generalizations of (real parts of) eigenvalues and eigenspaces as a starting point for a linear algebra for classes of time-varying linear systems, namely periodic, random, and perturbed (or controlled) systems. The book presents for the first time in one volume a unified approach via Lyapunov exponents to detailed proofs of Floquet theory, of the properties of the Morse spectrum, and of the multiplicative ergodic theorem for products of random matrices. The main tools, chain recurrence and Morse decompositions, as well as classical ergodic theory are introduced in a way that makes the entire material accessible for beginning graduate students.

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Genre : Mathematics
Author : Fritz Colonius
Publisher : American Mathematical Society
Release : 2014-10-03
File : 302 Pages
ISBN-13 : 9780821883198


Dynamical Systems

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Several distinctive aspects make Dynamical Systems unique, including: treating the subject from a mathematical perspective with the proofs of most of the results included providing a careful review of background materials introducing ideas through examples and at a level accessible to a beginning graduate student

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Genre : Mathematics
Author : Clark Robinson
Publisher : CRC Press
Release : 1998-11-17
File : 522 Pages
ISBN-13 : 9781482227871


Differential Equations Dynamical Systems And Linear Algebra

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This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.

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Genre : Mathematics
Author : Morris W. Hirsch
Publisher : Academic Press
Release : 1974-06-28
File : 373 Pages
ISBN-13 : 9780080873763


Handbook Of Linear Algebra Second Edition

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With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters. New to the Second Edition Separate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of matrices, generalized inverses, matrices over finite fields, invariant subspaces, representations of quivers, and spectral sets New chapters on combinatorial matrix theory topics, such as tournaments, the minimum rank problem, and spectral graph theory, as well as numerical linear algebra topics, including algorithms for structured matrix computations, stability of structured matrix computations, and nonlinear eigenvalue problems More chapters on applications of linear algebra, including epidemiology and quantum error correction New chapter on using the free and open source software system Sage for linear algebra Additional sections in the chapters on sign pattern matrices and applications to geometry Conjectures and open problems in most chapters on advanced topics Highly praised as a valuable resource for anyone who uses linear algebra, the first edition covered virtually all aspects of linear algebra and its applications. This edition continues to encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, and applications of linear algebra to various disciplines while also covering up-to-date software packages for linear algebra computations.

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Genre : Mathematics
Author : Leslie Hogben
Publisher : CRC Press
Release : 2013-11-26
File : 1906 Pages
ISBN-13 : 9781466507289


Dynamical Systems

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Dynamic tools of analysis and modelling are increasingly used in Economics and Biology and have become more and more sophisticated in recent years, to the point where the general students without training in Dynamic Systems (DS) would be at a loss. No doubt they are referred to the original sources of mathematical theorems used in the various proofs, but the level of mathematics is generally beyond them. Students are thus left with the burden of somehow understanding advanced mathematics by themselves, with· very little help. It is to these general students, equipped only with a modest background of Calculus and Matrix Algebra that this book is dedicated. It aims at providing them with a fairly comprehensive box of dynamical tools they are expected to have at their disposal. The first three Chapters start with the most elementary notions of first and second order Differential and Difference Equations. For these, no matrix theory and hardly any calculus are needed. Then, before embarking on linear and nonlinear DS, a review of some Linear Algebra in Chapter 4 provides the bulk of matrix theory required for the study of later Chapters. Systems of Linear Differ ential Equations (Ch. 5) and Difference Equations (Ch. 6) then follow to provide students with a good background in linear DS, necessary for the subsequent study of nonlinear systems. Linear Algebra, reviewed in Ch. 4, is used freely in these and subsequent chapters to save space and time.

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Genre : Business & Economics
Author : Pierre N.V. Tu
Publisher : Springer Science & Business Media
Release : 2013-11-11
File : 257 Pages
ISBN-13 : 9783662027790


Integrability Of Dynamical Systems Algebra And Analysis

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This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

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Genre : Mathematics
Author : Xiang Zhang
Publisher : Springer
Release : 2017-03-30
File : 390 Pages
ISBN-13 : 9789811042263


Dynamical Systems

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Chaos is the idea that a system will produce very different long-term behaviors when the initial conditions are perturbed only slightly. Chaos is used for novel, time- or energy-critical interdisciplinary applications. Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, as well as military applications, etc. This book presents the latest investigations in the theory of chaotic systems and their dynamics. The book covers some theoretical aspects of the subject arising in the study of both discrete and continuous-time chaotic dynamical systems. This book presents the state-of-the-art of the more advanced studies of chaotic dynamical systems.

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Genre : Mathematics
Author : Zeraoulia Elhadj
Publisher : CRC Press
Release : 2019-01-21
File : 307 Pages
ISBN-13 : 9780429647420


Model Reduction Of Complex Dynamical Systems

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This contributed volume presents some of the latest research related to model order reduction of complex dynamical systems with a focus on time-dependent problems. Chapters are written by leading researchers and users of model order reduction techniques and are based on presentations given at the 2019 edition of the workshop series Model Reduction of Complex Dynamical Systems – MODRED, held at the University of Graz in Austria. The topics considered can be divided into five categories: system-theoretic methods, such as balanced truncation, Hankel norm approximation, and reduced-basis methods; data-driven methods, including Loewner matrix and pencil-based approaches, dynamic mode decomposition, and kernel-based methods; surrogate modeling for design and optimization, with special emphasis on control and data assimilation; model reduction methods in applications, such as control and network systems, computational electromagnetics, structural mechanics, and fluid dynamics; and model order reduction software packages and benchmarks. This volume will be an ideal resource for graduate students and researchers in all areas of model reduction, as well as those working in applied mathematics and theoretical informatics.

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Genre : Mathematics
Author : Peter Benner
Publisher : Springer Nature
Release : 2021-08-26
File : 415 Pages
ISBN-13 : 9783030729837


Dynamical Systems Generated By Linear Maps

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The book deals with dynamical systems, generated by linear mappings of finite dimensional spaces and their applications. These systems have a relatively simple structure from the point of view of the modern dynamical systems theory. However, for the dynamical systems of this sort, it is possible to obtain explicit answers to specific questions being useful in applications. The considered problems are natural and look rather simple, but in reality in the course of investigation, they confront users with plenty of subtle questions and their detailed analysis needs a substantial effort. The problems arising are related to linear algebra and dynamical systems theory, and therefore, the book can be considered as a natural amplification, refinement and supplement to linear algebra and dynamical systems theory textbooks.

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Genre : Mathematics
Author : Ćemal B. Dolićanin
Publisher : Springer
Release : 2014-07-19
File : 205 Pages
ISBN-13 : 9783319082288


Network Based Analysis Of Dynamical Systems

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This book explores the key idea that the dynamical properties of complex systems can be determined by effectively calculating specific structural features using network science-based analysis. Furthermore, it argues that certain dynamical behaviours can stem from the existence of specific motifs in the network representation. Over the last decade, network science has become a widely applied methodology for the analysis of dynamical systems. Representing the system as a mathematical graph allows several network-based methods to be applied, and centrality and clustering measures to be calculated in order to characterise and describe the behaviours of dynamical systems. The applicability of the algorithms developed here is presented in the form of well-known benchmark examples. The algorithms are supported by more than 50 figures and more than 170 references; taken together, they provide a good overview of the current state of network science-based analysis of dynamical systems, and suggest further reading material for researchers and students alike. The files for the proposed toolbox can be downloaded from a corresponding website.

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Genre : Computers
Author : Dániel Leitold
Publisher : Springer Nature
Release : 2020-01-13
File : 119 Pages
ISBN-13 : 9783030364724