1) Phenomenology of Area-Preserving Maps
2) Symplectic Maps and KAM Theory
3) Twist Maps and Generating Functions
4) Birkhoff’s theorem and Periodic Orbits
5) Mather’s Variational Theorem and Cantori
6) Transport and Turnstiles

In this course we will review a classical problem in perturbation theory. We will consider a system having some hyperbolic manifold (point, periodic orbit, torus etc) with stable and unstable manifolds which coincide and study how they split when perturbing the system. In the regular case, we will review the well established  method, known as Melnikov-Poincaré method, which provides an asymptotic formula which measures the difference of the manifolds in a suitable section in terms of the perturbing parameter.

We will study the so-called singular case, where the system is degenerate and this formula is not always valid. We will focus in analytic systems where the splitting is exponentially small in the
perturbing parameter.

We will show some famous examples where this phenomenon occurs like the Hopf zero-singularity, the restricted three body problem or the problem of Arnold diffusion in the so-called a priori-stable systems.

Then we will give some ideas of how can prove a formula which gives the asymptotic measure of the splitting of separatrices in these cases.