An algorithm for searching hidden oscillations in dynamic systems is developed to help solve the Aizerman's, Kalman's and Markus-Yamabe's conjectures well-known in control theory. The first step of the algorithm consists in applying modified harmonic linearization methods. A strict mathematical substantiation of these methods is given using special Poincare maps. Subsequent steps of the proposed algorithms rely on the modern applied theory of bifurcations and numerical methods of solving differential equations. These algorithms help find and localize hidden strange attractors (i.e., such that a basin of attraction of which does not contain neighborhoods of equilibria), as well as hidden periodic oscillations. One of these algorithms is used here to discover, for the first time, a hidden strange attractor in the dynamic system describing a nonlinear Chua's circuit, viz. an electronic circuit with nonlinear feedback.