We will discuss an asymptotic behaviour of the set of occupational measures generated by a non-linear controlled dynamical system, and we will show that, as the time horizon goes to infinity, this set is approaching to a convex and compact set characterized by linear constraints. We will use the fact of such a convergence for analysis and solution of long-run average problems of optimal control and for approximating of the slow dynamics of singularly perturbed control systems. Theoretical developments will be illustrated by numerical examples. The presentation will be based on new results as well as on earlier results published in  V. Gaitsgory, On Representation of the Limit Occupational Measures Set of a Control Systems with Applications to Singularly Perturbed Control Systems", SIAM J. Control and Optimization, 43 (2004), No 1, pp 325-340.  V. Gaitsgory and S. Rossomakhine, Linear Programming Approach to Deterministic Long Run Average Problems of Optimal Control", SIAM J. Control and Optimization, 44 (2005/2006), No 6, pp. 2006-2037.