Many coexisting attractors, a case study of the almost-conservative H\'enon map

For dynamical systems in the plane, there can be many periodic attractors coexisting in a bounded region. They become easier to find in systems with small dissipation, which we call ``almost-conservative''. We ask what happens when there are many periodic attractors. That is the vague question we start with. For a test study, we chose the H\'enon map with a tiny dissipation. We tuned the other parameter to yield a case with 50 attracting periodic orbits. They have a total of 4259 periodic points. We describe how these orbits can be organized into families. In addition to two low-period orbits, the remaining 48 orbits can be classified into three families, which we describe in detail.