On the periodic oscillations of x¨=g(x)

The local isochronism of the periodic oscillations of x¨=g(x) (g(x)=−xf(x), f(0)>0, so that x=0 is a center) is equivalent to the Lyapunov stability of (0,0) for the system: x¨=g(x), y¨=yg′(x). The generalization of this property for the system x¨=g(x), y¨=y[g′(x)+3αg(x)(1+αx)−1], g′(0)<0, is studied by introducing an "artificial time'' τ, τ(t,x0,x˙0)=∫t0m(x(ξ,x0,x˙0))dξ, where m is a strictly positive continuous map and x a periodic solution, and by defining a general m-isochronism relative to this "time''; e.g., the concept of m-isochronous center is used. In particular, f-isochronism and (1+αx)−2-isochronism are studied and applied.