This paper proposes two non parametric tests for stationarity and white noise against the alternative of time-varying covariance structure with an application to euro/us dollar exchange rate. These tests are based on stability of evolutionary spectral density of the process. Graphical methods using the size and power, confirm the efficiency of our approach when compared with other stationarity tests. For the euro/us dollar exchange rate, the method reveals a break point and a short-term instability while the long-term stability is not rejected by the tests.