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Bifurcation theory and PDEs

Course Type: 
PhD Course
Master Course
Anno (LM): 
Second Year
Academic Year: 
October - January
50 h
CFU (LM): 

This course deals with bifurcation theory and applications to dynamical systems and PDEs, like the Lyapunov center theorem, Hopf bifurcation, traveling and Stokes waves for fluids. At the beginning we shall present the differential calculus and the implicit function theorem in Banach spaces. At the end I will deal also with the cases in which the classical implicit function theorem can not be applied since the linearized operator has an unbounded inverse and a version of the Nash-Moser implicit function theorem. 

  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
  7. Bifurcation in the variational setting
  8. Applications to bifurcation of periodic orbits,
  9. Hamiltonian and reversible systems, three body problem
  10. Hopf bifurcation,
  11. Lyapunov center Theorem
  12. Water waves, traveling waves,
  13. Benard Problem,
  14. A Nash-Moser Implicit function theorem
  15. Application to small divisors problem.
A-136 on Thursdays and A-134 on Fridays
Next Lectures: 

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