In this work quantifying the uncertainty associated with time dependent physical systems involving random parameters is explored. The simulation of such systems requires the basis dimension and computational resources for long time integration. This paper proposed the development of a computational approach based on a reformulation of the problem. Two variants of this approach are considered and evaluated. The first relies on an explicit expression to some distance between the corresponding solution on a reference deterministic system. The second variant achieves the same objective by concepts from optimal control theory, and yields more precise estimates at the price of a higher CPU cost, that is more appropriate for uncertain systems. As examples, the method applied to the case of a linear oscillator with uncertain properties, and to a stiff nonlinear chemical system involving uncertain reaction constants or rates. The tests show that the effectiveness of the proposed approaches, where the topology of the limit cycle does not change when the uncertain system parameters vary.