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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 inthe 2d plane. The vortices are considered as effective particles and theirinteraction can be desribed in classical mechanics terms. We first construct aPoincar\'e section, for a typical value of the energy, in order to acquire apicture of the structure of the phase space of the system. We divide the phasespace in different regions which correspond to qualitatively distinct motionsand we demonstrate its different temporal evolution in the "real" vortex-space.Our main emphasis is on the leapfrogging periodic orbit, around which weidentify a region that we term the "leapfrogging envelope" which involvesmostly regular motions, such as higher order periodic and quasi-periodicsolutions. We also identify the chaotic region of the phase plane surroundingthe leapfrogging envelope as well as the so-called walkabout and braidingmotions. Varying the energy as our control parameter, we construct abifurcation tree of the main leapfrogging solution and its instabilities, aswell as the instabilities of its daughter branches. We identify thesymmetry-breaking instability of the leapfrogging solution (in line withearlier works), and also obtain the corresponding asymmetric branches ofperiodic solutions. We then characterize their own instabilities (includingperiod doubling ones) and bifurcations in an effort to provide a moresystematic perspective towards the types of motions available to this dynamicalsystem.

A Hamiltonian Bifurcation Perspective on Two Interacting Vortex Pairs: From Symmetric to Asymmetric Leapfrogging, Period Doubling and Chaos