Summary
By following the bifurcation sequences of two main families of periodic orbits of the Hamiltonian\(H(\alpha ) = (\dot x^2 + \dot y^2 + x^2 y^2 )/2 + \alpha (x^2 + y^2 )/2\) as α→0, we show that they all destabilize in a systematic way (mainly by period-doubling bifurcations), and are unstable at α=0, suggesting that there is no stable periodic orbit in that limit. Still, despite this, and related results by other authors, it has not been rigorously proved to date that the HamiltonianH(0) is completely chaotic,i.e. that all of its periodic orbits are unstable.
Riassunto
Seguendo le sequenze di biforcazione di due principali famiglie di orbite periodiche dell’Hamiltoniana\(H(\alpha ) = (\dot x^2 + \dot y^2 + x^2 y^2 )/2 + \alpha (x^2 + y^2 )/2\) mentre α→0, si mostra che tutte destabilizzano in un modo sistematico (principalmente per biforcazioni a raddoppio di periodo) e sono instabili a α=0, il che suggerisce che non c’è nessuna orbita periodica stabile in quel limite. Tuttavia, nonostante ciò e i risultati riportati da altri autori non è stato rigorosamente provato fino ad ora che l’hamiltonianaH(0) è completamente caotica, cioè che tutte le sue orbite periodiche sono instabili.
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Sohos, G., Bountis, T. & Polymilis, H. Is the Hamiltonian\(H = (\dot x^2 + \dot y^2 + x^2 y^2 )/2\) completely chaotic?completely chaotic?. Nuov Cim B 104, 339–352 (1989). https://doi.org/10.1007/BF02728404
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DOI: https://doi.org/10.1007/BF02728404