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Monotonicity conditions for the period function of some planar Hamiltonian systems. (English) Zbl 0872.34026

Consider the differential equation \((*)\) \(x''+g(x)=0\) under the conditions (i) \(g\) is sufficiently smooth; (ii) \(g(0)=0\), \(xg(x)>0\) for \(x\neq 0\); (iii) there are two real numbers \(a<0<b\), such that \(G(a)= G(b)= \overline{c}\), \(G(x)<\overline{c}\) for \(x\in(a,b)\) where \(G\) is the primitive of \(g\). Then there exists a continuous family of periodic orbits \(y(c)\) with energy \(y^2/2+ G(x)=c< \overline{c}\). Let \(T(c)\) be the corresponding minimal period. The authors give new sufficient conditions for \(T(c)\) to be monotone which are less restrictive than some of the known conditions and easier to verify.

MSC:

34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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