  In this work we study the dynamical behavior of two interacting vortex pairs,
each one of them consisting of two point vortices with opposite circulation in
the 2d plane. The vortices are considered as effective particles and their
interaction can be desribed in classical mechanics terms. We first construct a
Poincar\'e section, for a typical value of the energy, in order to acquire a
picture of the structure of the phase space of the system. We divide the phase
space in different regions which correspond to qualitatively distinct motions
and we demonstrate its different temporal evolution in the "real" vortex-space.
Our main emphasis is on the leapfrogging periodic orbit, around which we
identify a region that we term the "leapfrogging envelope" which involves
mostly regular motions, such as higher order periodic and quasi-periodic
solutions. We also identify the chaotic region of the phase plane surrounding
the leapfrogging envelope as well as the so-called walkabout and braiding
motions. Varying the energy as our control parameter, we construct a
bifurcation tree of the main leapfrogging solution and its instabilities, as
well as the instabilities of its daughter branches. We identify the
symmetry-breaking instability of the leapfrogging solution (in line with
earlier works), and also obtain the corresponding asymmetric branches of
periodic solutions. We then characterize their own instabilities (including
period doubling ones) and bifurcations in an effort to provide a more
systematic perspective towards the types of motions available to this dynamical
system.
