Shadowing orbits for transition chains of invariant tori alternating with Birkhoff zones of instability

We consider a dynamical system that exhibits transition chains of invariant tori alternating with Birkhoff zones of instability in a two-dimensional centre manifold. It is known that there exist orbits that shadow the transition chains. It is also known that there exist orbits that cross the Birkhoff zones of instability. We describe a topological mechanism that allows one to join together the two types of orbits. We prove the existence of diffusing orbits that shadow the transition chains of invariant tori and cross the Birkhoff zones of instability. This approach is motivated by what has been called the large gap problem in Hamiltonian systems.