A simple procedure for deriving a uniform asymptotic expansion for the limit cycle in the vicinity of the Hopf bifurcation point for a two dimensional reaction system \[ u_{t} =D_{u}\Delta u+f\left( u,v;a\right) , \] \[ v_{t} =D_{v}\Delta v+g\left( u,v;a\right) \tag{b} \] is suggested. First, an algorithm allowing reduction of the system (ref {b}) to a second order differential equation representing a weakly nonlinear oscillator in the normal form is presented. Then, the Krylov-Bogoliubov-Mitropolski averaging method is applied to determine the appearance of subcritical or supercritical Hopf bifurcations. Using the asymptotic expansion for the limit cycle, appropriate normal modes are found. These are used for studying the appearance of Turing instabilities of the stable limit cycle. The authors conclude that the limit cycle may generate Turing instabilities in two ways. Namely, the amplified function is a product of a spatial eigenfunction and a time-periodic function either with the average close to one, or with the average zero. Therefore, for weak instabilities, one observes superposition of dominant inhomogeneous steady patterns with slight time-periodic oscillations with the frequency coinciding with that of the limit cycle. On the other hand, strong instabilities are characterized by intermittent switching between the inhomogeneous pattern represented by the set on which the spatial eigenfunction takes on positive values and a complementary pattern associated with the set on which the eigenfunction takes on negative values, thus producing the effect known as twinkling patterns. The frequency of these oscillations differs from the frequency associated with the limit cycle.