Dynamics and Bifurcations

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.

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

Review

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