Hyperbolic Dynamics

Recommended Prerequisites

Besides basic knowledge due to a bachelor degree in Mathematics, acquaintance in Differentiable Manifolds and Topology. 

Teaching Methods

Oral and written presentations in blackboard or projections.

Learning Outcomes

Students must be able to recognize, both in differential equations and in difference equations, the properties of hyperbolic dynamical systems, and to  know some of the most important results on their geometry, local or global stability of invariant subsets and structural stability. 

Work Placement(s)

No

Syllabus

- Dynamical Systems: Vector fields; Differential equations; Local flow; Discretisation; Difference equations; Suspension.

- Invariant sets; Limit sets; Transitivity; Structural stability.

- Local behaviour: Local stability; Invariant manifolds; Homoclinic Points.

- Global dynamics: global invariant manifolds.

- Examples (some of these): Linear dynamics; Symbolic dynamics; Smale Horseshoe; Axiom A; Anosov diffeomorphisms; Kupka-Smale diffeomorphisms; Morse-Smale difeomorphisms.

Head Lecturer(s)

José Ferreira Alves

Assessment Methods

Assessment
Frequency: 50.0%
Exam: 50.0%

Bibliography

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.

V.I. Arnold, Ordinary Differential Equations, MIT press, 1973.

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.

M. Hirsch, S. Smale, and R.L. Devaney, Differential Equations, Dynamical Systems, and Introduction to Chaos, Elsevier, 2004.

M. Irwin, Smooth Dynamical Systems, Academic Press, London, 1980.

W. de Melo and S. Van Strien, One-dimensional Dynamics, Springer-Verlag, Berlin, 1993.

Z. Nitecki, Differentiable Dynamics: An introduction to the Orbit Structure of Dipheomorphisms, MIT Press, Cambridge, 1971.

J. Palis, Jr., and W. de Melo, Geometric Theory of Dynamical Systems: an introduction, Springer-Verlag, New York, 1982.

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1999.

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987.