Introduction to Dynamical Systems - Part 1/2 (J. Fejoz)

1. Bibliography

M. Brin and G. Stuck, Introduction to Dynamical Systems, 2002 (PDF file)
A good part of the material of the first part of the course is contained in the first chapter of this book.
V. Arnold, Ordinary Differential Equations, 1992 (DJVU file)
Excellent introductory book on ODEs
V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 1988 (DJVU file)
Advanced book with a wealth of topics studied. A jewel!
V. Arnold, Mathematical Methods of Classical Mechanics, 1989 (DJVU file)
J. Milnor, Morse Theory, 1969 (PDF file)

September 18: Notion of dynamical system. Conjugacy. Example of the pendulum
September 26: Rotations of the circle with rational or irrational rotation number. Generalization to rotations of \(𝕋^n\) (resonance, ergodicity, first instance of Birkhoff ergodic theorem)
October 3: An expanding endomorphism of the circle. Semi-conjugacy to the shift
October 10: A hyperbolic toral automorphism. Structural stability (Anosov theorem: see Arnold, Geometrical Methods, chap. 3)
October 17: Hamiltonian systems, integrability
For an excellent informal introduction to differential forms see Arnold, Mathematical Methods of Classical Mechanics.
October 24: Perturbation theory, normal form along a Diophantine invariant torus (§ 19 of Arnold, Geometrical Methods)
Wednesday January 8, 3-5 pm: Exam