Approximate calculation of amplitudes of cycles bifurcating in the presence of resonances

The procedure of approximate calculation of amplitudes for periodic solutions bifurcating from rest points in the presence of resonance is studied for a class of dynamical systems. This class includes equations of spring beam oscillations located on elastic foundations, autonomous systems of ordinary differential equations, hydrodynamical systems etc. The methodological basis of the procedure is the Lyapunov–Schmidt method considered in the context of general theory of smooth $SO(2)-$equivariant Fredholm equations (in infinite dimensional Banach spaces). The topic of the paper develops and extends the earlier research of B. M Darinsky, Y. I. Sapronov, and V. A. Smolyanov.