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Geometrical aspects of stability theory for Hill’s equations. (English) Zbl 0840.34047

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.
Reviewer: L.Hatvani (Szeged)

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
70J40 Parametric resonances in linear vibration theory
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