KNOWLEDGE AND UNDERSTANDING
By the end of the course the student is expected to be familiar with fundamental ideas of nonlinear analysis, bifurcation theory
with applications in dynamical systems and PDEs.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING By the end of the course the student is expected to be able to apply methods and techniques of nonlinear analysis acquired to solve problems and exercises of medium difficulty. The exercises can also be proposed as easy theoretical results.
By the end of the course the student is expected to be able to recognize and apply basic techniques of nonlinear analysys and also to recognize situations and problems in which these techniques can be used advantageously.
By the end of the course the student is expected to be able to express himself with proficient command of language and exposure security on the topics of the course.
By the end of the course the student is expected to be able to consult standard texts of nonlinear analysis and bifurcation theory.
Differential and integral calculus for functions of several variables, Basic facts on Banach and Hilbert spaces, Existence, uniqueness and smooth dependence one the data theorem for solutions of the Cauchy problem for ordinary differential equations
1. Differential calculus in Banach spaces, Frechet and Gateuax derivatives, Taylor formula,
2. Analytic functions
3. Local inversion and implicit function theorems in Banach spaces
4. Lyapunov-Schmidt reduction
5. Bifurcation theorem from the simple eigenvalue
6. A bifurcation theorem from a multiple eigenvalue
6. Bifurcation in the variational setting
7. Applications to bifurcation of periodic orbits,
8. Hamiltonian and reversible systems, three body problem
9. Hopf bifurcation,
10. Lyapunov center Theorem
11. Water waves, traveling waves,
12. Benard Problem,
13. A Nash-Moser Implicit function theorem
14. Application to small divisors problem.
Lectures and exercise sessions
Written and oral exam. Written exam: 2 hours, 2 exercises. Oral discussion.
A. Ambrosetti. G. Prodi “A primer of nonlinear analysis”,
J. Moser, E. Zehnder “Notes on Dynamical Systems”
Nirenberg: “Topics in nonlinear functional analysis”