LAUSR

In this work, we consider the Kepler problem with a family of singular dissipations of the form $$-\frac{k}{|x|^\beta }\dot{x},\quad k>0, \beta >0.$$ - k | x | β x ˙ , k > 0 , β > 0 . We present some results about the qualitative dynamics as $$\beta $$ β increases from zero (linear drag) to infinity. In particular, we detect some threshold values of $$\beta $$ β , for which qualitative changes in the global dynamics occur. In the case $$\beta =2$$ β = 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.