Invariant manifolds course

Schedule

The course (SMA5940) runs during the second semester of 2016. Lecture hours are Tuesdays and Thursdays between 2–4pm in room 5-002. On Tuesdays there will be lectures and on Thursdays discussions of the exercises.

• Tue 9-8: lecture on stability
• Thu 11-8: exercises: 2.6, 2.7
• Tue 16-8: lecture on hyperbolicity and (un)stable manifolds
• Thu 18-8: hand in: 2.8, 2.10; exercises: 2.9, 3.3 (and start lecture on Perron method proof)
• Tue 23-8: lecture on the Perron method proof
• Thu 25-8: hand in: 3.5; exercises: 3.6, 3.8
• Tue 30-8: no lecture today
• Thu 1-9: hand in: 3.10, 3.11; guest lecture on inertial manifolds for PDEs by Prof. Sergey Zelik, at 14:00–15:30h in room 4-111
• Fri   2-9: extra guest lecture on Blue-sky catastrophy in slow-fast systems (part 1) by Prof. Dmitry Turaev, at 14:00–15:00h in room 3-011
• Mon 5-9: extra guest lecture on Blue-sky catastrophy in slow-fast systems (part 2) by Prof. Dmitry Turaev, at 10:00–11:00h in room 4-001
• Tue 6-9: no lectures this week
• Thu 8-9: no lectures this week
• Tue 13-9: lecture: dependence on parameters, non-autonomous systems
• Thu 15-9: hand in: 3.12, exercises: 3.13
• Tue 20-9: lecture: introduction of normal hyperbolicity
• Thu 22-9: hand in: none, exercises: 3.15
• Tue 27-9: lecture: the graph transform
• Thu 29-9: hand in: 4.7, exercises: 4.8
• Tue 4-10: no lectures this week
• Thu 6-10: no lectures this week
• Tue 11-10: lecture: Andronov–Hopf bifurcation
• Thu 13-10: hand in: 5.1, lecture: Andronov–Hopf bifurcation
• Tue 18-10: lecture: Andronov–Hopf bifurcation
• Thu 20-10: lecture: isochrons and phase response in limit cycles
• Tue 25-10: lecture: proof of persistence of NHIMs
• Thu 27-10: lecture: proof of persistence of NHIMs
• Tue 1-11: lecture: proof of persistence of NHIMs, fiber contraction theorem
• Thu 3-11: no lecture today
• Tue 8-11: lecture: smoothness of the persistent NHIM
• Thu 11-11: hand-in: 5.4, 5.5, lecture: wrap-up persistence of NHIMs
• Tue 15-11: no lecture today
• Thu 17-11: hand in: 5.9; exercises: 5.8
• Tue 22-11: lecture: (smooth) invariant foliations
• Thu 17-11: last meeting, wrap-up

Literature

As literature we use two books:

• "Ordinary Differential Equations with Applications" by Carmen Chicone, Springer Texts in Applied Mathematics 34
• "Normally hyperbolic invariant manifolds in dynamical systems" by Stephen Wiggins, Springer Applied Mathematical Sciences 105

Further material and exercises will be provided in separate lecture notes that I will update during the course.
Separately available is my translation in English of Hadamard's original graph transform article (pages 224–228).
Also still available are the notes from the preliminary lectures.