A First Course in Dynamics
with a Panorama of Recent Developments

An advanced undergraduate introduction combining mathematical rigour with copious examples of important applications.

Boris Hasselblatt (Author), Anatole Katok (Author)

9780521587501, Cambridge University Press

Paperback, published 23 June 2003

436 pages, 140 b/w illus.
25.1 x 17.7 x 2 cm, 0.758 kg

'I would recommend that all senior undergraduate physics and engineering students should familiarize themselves with its contents … it is a recommendation for a mathematical perspective on dynamics from an engineer for engineers - and from my discussions with practical physicists a similar recommendation from them.' Contemporary Physics

The theory of dynamical systems is a major mathematical discipline closely intertwined with all main areas of mathematics. It has greatly stimulated research in many sciences and given rise to the vast new area variously called applied dynamics, nonlinear science, or chaos theory. This introduction for senior undergraduate and beginning graduate students of mathematics, physics, and engineering combines mathematical rigor with copious examples of important applications. It covers the central topological and probabilistic notions in dynamics ranging from Newtonian mechanics to coding theory. Readers need not be familiar with manifolds or measure theory; the only prerequisite is a basic undergraduate analysis course. The authors begin by describing the wide array of scientific and mathematical questions that dynamics can address. They then use a progression of examples to present the concepts and tools for describing asymptotic behavior in dynamical systems, gradually increasing the level of complexity. The final chapters introduce modern developments and applications of dynamics. Subjects include contractions, logistic maps, equidistribution, symbolic dynamics, mechanics, hyperbolic dynamics, strange attractors, twist maps, and KAM-theory.

1. What is a dynamical system?
Part I. Simple Behavior in Dynamical Systems: 2. Systems with stable asymptotic behavior
3. Linear maps and linear differential equations
Part II. Complicated Behavior in Dynamical Systems: 4. Quasiperiodicity and uniform distribution on the circle
5. Quasiperiodicity and uniform distribution in higher dimension
6. Conservative systems
7. Simple systems with complicated orbit structure
8. Entropy and chaos
9. Simple dynamics as a tool
Part III. Panorama of Dynamical Systems: 10. Hyperbolic dynamics
11. Quadratic maps
12. Homoclinic tangles
13. Strange attractors
14. Diophantine approximation and applications of dynamics to number theory
15. Variational methods, twist maps, and closed geodesics
Appendix
Solutions.

Subject Areas: Nonlinear science [PBWR], Applied mathematics [PBW]