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.

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

ZAMPIERI, Gaetano
1989-01-01

Abstract

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.
1989
Isochronism, generalized isochronism, Lyapunov stability
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/393343
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 10
  • ???jsp.display-item.citation.isi??? 7
social impact