5. Heterogeneous solutions

The search for heteroclinic solutions using the variational method is mainly based on the direct method of calculating variations, i.e. on minimizing the action functional. Arguments are therefore expected to be simpler than in the search for homoclinic solutions. However, other problems arise, notably in the choice of the class of functions on which to define the action functional. In particular, the choice of topology over this class of functions is sometimes tricky.

In this paragraph, we consider only second-order Lagrangian systems, the case of Hamiltonian systems being technically much more complicated and having been much less studied.