Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

In the current paper, the problem of sextic anharmonic oscillators is investigated. There are three integrable cases of this problem. Emphasis is placed on two integrable cases, and a full description of each one is provided. The separated functions of the first and second integrability cases are transformed from a higher degree to the third and fourth degrees. Respectively, the periodic solution is obtained using Jacobi’s elliptic functions. The topology of phase space and Liouville tori’s bifurcations are discussed. The phase portrait is studied to determine singular points and classify their types in addition to the graphic representation for each of them. Finally, the numerical illustrations are introduced using the Poincaré surface section to emphasize the problem’s integrability.