Synopses & Reviews
In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.
J.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations "This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples."--MATHEMATIKA
The subject of differential and difference equations is an old and much-honored chapter in science, one which germinated in applied fields such as celestial mechanics, nonlinear oscillations, and fluid dynamics. In recent years, due primarily to the proliferation of computers, dynamical systems has once more turned to its roots in applications with perhaps a more mature look. Many of the available books and expository narratives either require extensive mathematical preparation, or are not designed to be used as textbooks. The authors have filled this void with the present book.
This comprehensive textbook is designed to take undergraduate and beginning graduate students of mathematics, science, and engineering from the rudimentary beginnings to the exciting frontiers of dynamical systems and their applications. It is a masterful exposition of the foundations of ordinary differential and difference equations from the contemporary viewpoint of dynamical systems and bifurcations. In both conception and execution, the authors implemented a fresh approach to mathematical narration. Fundamental ideas are explained in simple settings, the ramifications of theorems are explored for specific equations, and above all, the subject is related in the guise of a mathematical epic. With its insightful and engaging style, as well as its numerous computer-drawn illustrations of notable equations of theoretical and practical importance, this unique book will simply captivate the attention of students and instructors alike. 345 illustrations.
Table of Contents
Part I: Dimension One * Chapter 1. Scalar Autonomous Equations * Chapter 2. Elementary Bifurcations * Chapter 3. Scalar Maps * Part II: Dimension One and One Half * Chapter 4. Scalar Nonautonomous Equations * Chapter 5. Bifurcation of Periodic Equations * Chapter 6. On Tori and Circles * Part III: Dimension Two * Chapter 7. Planar Autonomous Systems * Chapter 8. Linear Systems * Chapter 9. Near Equilibria * Chapter 10. In the Presence of a Zero Eigenvalue * Chapter 11. In the Presence of Purely Imaginary Eigenvalues * Chapter 12. Periodic Orbits * Chapter 13. All Planar Things Considered * Chapter 14. Conservative and Gradient Systems * Chapter 15. Planar Maps * Part IV: Higher Dimensions * Chapter 16. Dimension Two and One Half * Chapter 17. Dimension Three * Chapter 18: Dimension Four * Farewell * Appendix: A Catalogue of Fundamental Theorems * References * Index