A dissipative Kepler problem with a family of singular drags

Abstract

In this work, we consider the Kepler problem with a family of singular dissipations of the form $$-\frac{k}{|x|^\beta }\dot{x},\quad k>0, \beta >0.$$ We present some results about the qualitative dynamics as $$\beta $$ increases from zero (linear drag) to infinity. In particular, we detect some threshold values of $$\beta $$, for which qualitative changes in the global dynamics occur. In the case $$\beta =2$$, we refine some results obtained by Diacu and prove that, unlike for the case of the linear drag, the asymptotic Runge–Lenz vector is discontinuous.