Geometrical aspects of stability theory for Hill's equations (Q1905658)

For the equation \(x''+ (a+ bp(t)) x= 0\), \(p(t)\equiv p(t+ 2\pi)\) (\(a\), \(b\) are real parameters) the authors introduce the Hill's map by \(H: (a, b)\mapsto P_{a, b}\), where \(P_{a, b}\) is the Poincaré (or period) matrix of the equation. They give a global geometric picture of this map, which explains the nature of the classical stability domains in the parameter plane.