Introduction to Applied Nonlinear Dynamical Systems and Chaos

This introduction to applied nonlinear dynamics and chaos places emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about their behavior. The new edition has been updated and extended throughout, and contains a detailed glossary of terms.

From the reviews:

“Will serve as one of the most eminent introductions to the geometric theory of dynamical systems.”

Table of Contents

1 Equilibrium Solutions, Stability, and Linearized Stability

2 Liapunov Functions

3 Invariant Manifolds: Linear and Nonlinear Systems

4 Periodic Orbits

5 Vector Fields Possessing an Integral

6 Index Theory

7 Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows

8 Asymptotic Behavior

9 The Poincar´e-Bendixson Theorem

10 Poincar´e Maps

11 Conjugacies of Maps, and Varying the Cross-Section

12 Structural Stability, Genericity, and Transversality

13 Lagrange’s Equations

14 Hamiltonian Vector Fields

15 Gradient Vector Fields

16 Reversible Dynamical Systems

17 Asymptotically Autonomous Vector Fields

18 Center Manifolds

19 Normal Forms

20 Bifurcation of Fixed Points of Vector Fields

21 Bifurcations of Fixed Points of Maps

22 On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution

23 The Smale Horseshoe

24 Symbolic Dynamics

25 The Conley–Moser Conditions, or “How to Prove That a Dynamical System is Chaotic”

26 Dynamics Near Homoclinic Points of

Two-Dimensional Maps

27 Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields

28 Melnikov’s Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields

29 Liapunov Exponents

30 Chaos and Strange Attractors

31 Hyperbolic Invariant Sets: A Chaotic Saddle

32 Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems

33 Global Bifurcations Arising from Local Codimension—TwoBifurcations

34 Glossary of Frequently Used Terms

About the Author

Stephen Ray Wiggins is a Cherokee-American applied mathematics researcher and educator, also of British heritage, best known for his contributions in nonlinear dynamics, chaos theory and nonlinear phenomena.