The investigation algorithm of stability of periodic oscillations in the problem for the Andronov–Hopf bifurcation

A new method for analysis of the stability of free oscillations under the conditions of the Andronov–Hopf bifurcation is suggested. In contrast to commonly applied methods, the algorithm suggested does not require the construction of integral varieties and the transition to normal forms. The algorithm is based on the comparison between the characters of stability of the stationary state of the system and the free oscillations being born. The method suggested enables us to simplify essentially the analysis of stability and obtain simple explicit criteria of stability and instability, and also define the type of bifurcation.