Hilbert–Arnold problem

In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of limit cycles. It asks whether in a generic finite-parameter family of smooth vector fields on a sphere with a compact parameter base, the number of limit cycles is uniformly bounded across all parameter values. The problem is historically related to Hilbert's sixteenth problem and was first formulated by Russian mathematicians Vladimir Arnold and Yulij Ilyashenko in the 1980s.

It's closely related to the "infinitesimal Hilbert's sixteenth problem", although they are not synonyms. In Arnold's Problems there are many questions related to the Hilbert–Arnold problem: 1978–6, 1979–16, 1980–1, 1983–11, 1989–17, 1990–24, 1990–25, 1994–51 and 1994–52.