It asks whether in a generic[disambiguation needed] 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.
While Hilbert's original question focused on polynomial vector fields, mathematical attention shifted toward properties of generic families[disambiguation needed] within certain classes.
Unlike polynomial systems, typical smooth systems on a sphere can have arbitrarily many hyperbolic limit cycles that persist under small perturbations.
However, the question of uniform boundedness across parameter families remains meaningful and forms the basis of the Hilbert–Arnold problem.
[2] Due to the compactness of both the parameter base and phase space, the Hilbert–Arnold problem can be reduced to a local problem studying bifurcations of special degenerate vector fields.
This leads to the concept of polycycles—cyclically ordered sets of singular points[disambiguation needed] connected by phase curve arcs—and their cyclicity, which measures the number of limit cycles born in bifurcations.
The local version of the Hilbert–Arnold problem asks whether the maximum cyclicity of nontrivial polycycles in generic k-parameter families (known as the bifurcation number
Ilyashenko and Yakovenko proved in 1995 that the elementary bifurcation number
[4] In 2003, mathematician Vadim Kaloshin established the explicit bound