Is chaos always suppressed parametrically?

Great success and profit of the chaos suppression phenomenon in applications led to the widespread opinion that chaotic oscillations may always be stabilized by parametric perturbations. Nevertheless, in what cases the chaos can be suppressed by such a manner? In general, this question means that we should perturb the system strictly within the chaoticity region, i.e. all perturbed parameters should not fall outside the limits of this region. In the present paper, by a Duffing system we construct an analytic example when parametric perturbations cannot lead to the suppression of chaos if they belong to the chaoticity region. Our analysis is based on the Melnikov method which gives us a criterion for the observation of chaos. The obtained results are in excellent agreement with numerical simulations.