Etude analytique du voisinage de la resonance 4:4:1 dans les systemes a trois degres de liberte

The periodic solutions for an Hamiltonian system with $$H = \frac{1}{2}\mathop \Sigma \limits_1^3 (\dot x_\alpha ^2 + \omega _\alpha ^2 x_\alpha ^2 ) - \varepsilon x_1 x_\alpha ^2 - \eta x_2 x_\alpha ^2 $$ are investigated analytically. The frequencies ω_α, α=1, 2, 3 are assumed near the ratio 4—4—1. We find different families of periodic solutions whose periods are in the vicinity of the period T′=2π/ω₃=2π/ω′. As in the case of the problem with two degrees of liberty, for particular values of ω₁, ω₂, ω₃ and ε, η, we find that the families near the x₃-axis are discontinuous. These families are periodic with periods near the period T′ in a region for ε, η, approximatively 0; 0.4] if we choose \(\omega ' = \sqrt {0.1} \) and h=0.00765.