ABSTRACT
In this chapter, the authors present evidence regarding the existence of irregular (chaotic) motion. They emphasize the association of this dynamical situation with the existence or creation of homoclinic orbits to a periodic orbit (meaning by this in most examples just a fixed point). The authors analyse the situation of a transverse homoclinic orbit showing that it is the limit of a family of the periodic orbits. They considers how these periodic orbits are created at a homoclinic tangency. The transverse heteroclinic crossings with the folding and accumulation processes produce secondary crossings between the stable and unstable manifolds of each periodic orbit hence producing (secondary) homoclinic orbits. The homoclinic tangency is a structurally unstable situation since the nearby maps (again, in a reasonably defined space of the maps) might have no manifold crossing at all or (in general) two transverse crossings.
