Differential Equations: A Dynamical Systems Approach: Higher-Dimensional Systems

ISBN-13: 9780387943770 / Angielski / Twarda / 1995 / 602 str.

John H. Hubbard; Beverly H. West

Opis

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, had led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math- ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface As in Part I, this book concentrates on understanding the behavior of dif- ferential equations, rather than on solving the equations. Part I focused on differential equations in one dimension; this volume attempts to understand differential equations in n dimensions. The existence and uniqueness theory carries over with almost no changes.

Spis treści

of Part II.- Systems of Ordinary Differential Equations The Higher-Dimensional Theory x? = f(t,x).- 6 Systems of Differential Equations.- 6.1 Graphical Representation of Systems.- 6.2 Theorems for Systems of Differential Equations.- 6.3 Example: Sharks and Sardines.- 6.4 Higher Order Equations.- 6.5 Mechanical Systems with One Degree of Freedom.- 6.6 Essential Size, Conservative Laws.- 6.7 The Two-Body Problem.- 6.8 Flows.- 6 Exercises.- 7 Systems of Linear Differential Equations.- 7.1 Linear Differential Equations in General.- 7.2 Linearity and Superposition Principles.- 7.3 Linear Differential Equations with Constant Coefficients: Eigenvectors and Decoupling.- 7.4 Linear Differential Equations with Constant Coefficients: Exponentials of Matrices.- 7.5 Two by Two Matrices and the Bifurcation Diagram.- 7.6 Eigenvalues and Global Behavior.- 7.7 Nonhomogeneous Linear Equations.- 7 Exorcises.- 8 Systems of Nonlinear Differential Equations.- 8.1 Zeroes of Vector Fields and Their Linearization.- 8.2 Sources are Sources and Sinks are Sinks.- 8.3 Saddles.- 8.4 Limit Cycles.- 8.5 The Poincaré-Bendixson Theorem.- 8.6 Symmetries and Volume-Preserving Equations.- 8.7 Chaos in Higher Dimensions.- 8.8 Structural Stability.- 8 Exercises.- 8* Structural Stability.- 8*.1 Preliminaries for Structural Stability.- 8*.2 Structural Stability of Sinks and Sources.- 8*.3 Time to Pass by a Saddle.- 8*.4 Structural Stability of Limit Cycles.- 8*.5 Why Poincaré-Bendixson Rules Out “Chaos” in the Plane.- 8*.6 Structural Stability in the Plane.- 8* Exercises.- 9 Bifurcations.- 9.1 Saddle-Node Bifurcation.- 9.2 Andronov-Hopf Bifurcations.- 9.3 Saddle Connections.- 9.4 Semistable Limit Cycles.- 9.5 Bifurcation in One-Parameter Families.- 9.6 Bifurcation in Two-Parameter Families.- 9.7 Grand Example.- 9 Exercises.- Appendix L: Linear Algebra.- L1 Theory of Linear Equations: In Practice.- L2 Theory of Linear Equations: Vocabulary.- L3 Vector Spaces and Inner Products.- L4 Linear Transformations and Inner Products.- L5 Determinants and Volumes.- L6 Eigenvalues and Eigenvectors.- L7 Finding Eigenvalues: The QR Method.- L8 Finding Eigenvalues: Jacobi’s Method.- Appendix L Exercises.- Appendix L Summary.- Appendix T: Key Theorems From Parts I and III.- References.- Answers to Selected Problems.