Bifurcations of symplectic maps with an application to the elliptic restricted three-body problem

Abstract Poincaré maps are discrete dynamical systems whose fixed points correspond to periodic orbits. When the dynamics is symplectic, Poincaré maps are symplectic on appropriate sections, and their linearization is a symplectic matrix. In this article we present an algebraic framework for the classification of all significant spectral types of symplectic maps. We explicitly determine, in the space of principal invariants of the linearization matrix, one algebraic variety and two semialgebraic varieties that correspond to specific changes in the spectral type of the equilibrium. The semialgebraic varieties are the decomposition of one algebraic variety (a discriminant locus) into parts associated to different spectral changes. The spectral changes are focus-center or focus-saddle depending on inequalities involving a polynomial expression obtained via a Sturm’s process. These varieties constitute the boundaries of a decomposition of the space of principal invariants in domains of fixed spectral type. We analyze such varieties thoroughly for generic 2D and 3D systems. When the dynamics depends on parameters, the decomposition can be pulled back in the space of physical parameters to obtain the bifurcation diagram for the system. We conclude the manuscript with an application to the elliptic restricted three-body problem. This application allows us to draw bifurcation diagrams that extend known results.