Entrainment of a limit cycle by a periodic external excitation is investigated with the Prigogine-Lefever-Nicolis model for chemical reaction. In the neighbourhood of the marginal point a quasi-harmonic theory is developed and the theoretical prediction is compared with the numerical computation. The stability of the entrained oscillation is examined in particular. The instabilities are classified into two types, i.e., hard- and soft-mode instabilities, by the use of the Floquet exponents which are, calculated by a non-perturbational method. The hard-mode instability corresponds to the limit of entrainment and the amplitude is subject to a modulation beyond the threshold. The frequency of modulation is estimated from the Floquet exponent, which is compared with the results of numerical computation. The soft-mode instabilitiy corresponds to a jump phenomenon in the amplitude. The smaller amplitude branch is liable to a modulation due to a superimposed hard-mode instability. As a whole, a reasonable agreement is obtained between numerical and theoretical results.