Outline:

Lecture 1. The paradox of the inverted pendulum and its explanations, topological and geometrical.

Lecture 2. The bicycle and the pendulum — a connection.

Lecture 3. More on the bike: Foote’s theorem, Menzin’s conjecture, equivalence with Hill’s equation.

Lecture 4. Rotating potentials and symplectic averaging.

Lecture 5. Gaussian curvature and gyroscopes.

Outline:

1.Introduction

1.1.Examples for time delayed systems

1.2.Basic properties 

2.Basic theory

2.1.Existence, uniqueness, continuous dependence

2.2.Continuation: forward, backward

2.3.Differentiability

3.Linear equations

3.1.Generator, spectrum

3.2.Decomposition of the phase space

3.3.Variation of constants formula

3.4.Linearization

4.Near equilibrium and periodic orbits

4.1.Saddle-point property

4.2.Local invariant manifolds: unstable, center, stable, center-unstable, center-stable

5.Delayed monotone feedback

5.1.Global attractor

5.2.Unstable sets of equilibria

5.3.Oscillation frequencies

5.4.Periodic orbits, Floquet multipliers

5.5.Connecting orbits

5.6.Graph representations

5.7.Morse decomposition of the global attractor

6.Equations with state-dependent delays

6.1.Basic theory

6.2.Smoothness problems

7.Problems

References

•H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.

•T. Krisztin, H.-O. Walther, J. Wu, Shape, Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive feedback, Fields Institute Monographs, AMS, 1999.

•O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations, Springer, 1995.

•J. K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, 1993.

•F. Hartung, T. Krisztin, H.-O. Walther, J. Wu, Functional differential equations with state-dependent delay: theory and applications, In: Handbook of differential equations: Ordinary differential equations. Vol. 3. Elsevier; North-Holland, 2006. pp. 435-545.