Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems are developed. This includes the harmonic balance method and the construction of counterexamples to Kalman's problem. An analytical criterion for the existence of periodic solutions is proved, such that the existence of hidden oscillations, a basin of attraction of which does not contain neighborhoods of equilibria, is detected. To construct a counterexample to Kalman's problem, the computational procedure is developed for the sequence of nonlinearities. The recursive construction of periodic solutions is found by replacing the nonlinearity by the strictly increasing function. The successive construction of periodic solutions for system is found by replacing the nonlinearity by the strictly increasing function.