Family 143Dynamical systems and ergodic theoryLeanNot yet digested
Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
Manuscripts in this family · 2
Uniform bounds for planar polynomial limit cycles
Sep 24, 2026·uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
Two limit cycles for quintic Liénard systems
Sep 24, 2026·two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdfLean
Every classical Liénard system , , with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.