  Let M = R n or possibly a Riemannian, non compact manifold. We consider
semi-excited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M)
induced by a non-degenerate periodic orbit $\gamma$ 0 of semi-hyperbolic type,
which is contained in the non critical energy surface {H 0 = 0}. By
semi-hyperbolic, we mean that the linearized Poincar{\'e} map dP 0 associated
with $\gamma$ 0 has at least one eigenvalue of modulus greater (or less) than
1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is
not an eigenvalue, which implies a family $\gamma$(E) with the same properties.
It is known that an infinite number of periodic orbits generally cluster near
$\gamma$ 0 , with periods approximately multiples of its primitive period. We
construct the monodromy and Grushin operator, adapting some arguments by
[NoSjZw], [SjZw], and compare with those obtained in [LouRo], which ignore the
additional orbits near $\gamma$ 0 , but still give the right quantization rule
for the family $\gamma$(E).
