On the periodic oscillations of \(\ddot x=g(x)\)
The paper is a continuation of the author's work in [J. Differ. Equations 74, No.2, 254-265 (1988; Zbl 0668.34051)]. For the scalar equation \(x''+xf(x)=0\), \(f(0)>0\), and for a strictly positive function m and for a periodic solution \(x(.,x_ 0,x'_ 0)\) the change in time is performed by the formula \(\tau (t,x_ 0,x'_ 0)=\int^{t}_{0}m(x(\theta,x_ 0,x'_ 0))d\theta.\) It is said that the origin is an m-isochronous center for the equation if all solutions in a neighbourhood of the origin have the same period with respect to the new time associated to m. Conditions for m-isochronism are given and it is proved that the functions f for which the origin is f-isochronous and m-isochronous for \(m(x)=(1+\alpha x)^{-2}\) are defined by \(f(x)=f(0)/(1+\alpha x)^ 3.\) A formula is obtained for functions f for which only m-isochronism with respect to the above m holds. It is proved that the origin is a stable equilibrium for the system \(x''+xf(x)=0\), \(y''+yf(x)=0\) iff \(f(0)>0\) and the origin is f-isochronous for the first equation in the system.
scientific article; zbMATH DE number 4148595
scientific article