In this talk, we first present a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and van der Pol Duffing equation whose noises are naturally degenerate. Other applications include stochastic May-Leonard system and random systems with infinitely many equivalent classes.