Hilbert–Arnold problem

In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of limit cycles of the dynamics of a flow on a sphere, and whether the number of such cycles can be bounded. A flow on a sphere means that you can imagine that the velocities of particles are prescribed, and a limit cycle is a limit of those velocities, like the Gulf Stream on the globe. The problem asks whether in a generic family of smooth vector fields, smoothly parameterized over a compact set in finite dimensional Euclidean space, the number of limit cycles is uniformly bounded across all parameter values. Thus, under perturbations of climate conditions, it asks whether there is a bounded number of "Gulf Streams". 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. 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.