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EnglishThe 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.
| Titolo: | On the periodic oscillations of x¨=g(x) |
| Autori: | |
| Data di pubblicazione: | 1989 |
| Rivista: | |
| 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. |
| Handle: | http://hdl.handle.net/11562/393343 |
| Appare nelle tipologie: | 01.01 Articolo in Rivista |
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