Differential Equations: A Dynamical Systems Approach Higher-Dimensional Systems

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.

of Part II.- Systems of Ordinary Differential Equations The Higher-Dimensional Theory x? = f(t,x).- 6 Systems of Differential Equations.- 7 Systems of Linear Differential Equations.- 8 Systems of Nonlinear Differential Equations.- 8* Structural Stability.- 9 Bifurcations.- 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.