A new method of analysis of degeneracies of arbitrary co-dimension in nonlinear systems of ODEs depending on parameters was suggested. The method is based on the use of variational equations of higher order, which are obtained recursively from the original equations and each other. These equations may be computed with the help of computer algebra systems. Degeneracies are understood in a broad sense. Bifurcations, the loss of stability, geometric peculiarities in the phase space (folds and pleats of the families of solutions) etc.\ may be effectively studied by this method. As an application, the equation of oscillations of the satellite in the plane of its elliptic orbit was considered, for which some new degenerate periodic solutions were found. Besides, the classical center-focus problem was considered. It was shown that all cases ever studied for this problem (i.e. systems with one edge of the Newton polygon) may be investigated by the method. Within this problem, perturbed Hamiltonian system with two edges of the Newton polygon were studied for the first time. Explicit formulas for the asymptotics of the Poincare mapping were obtained.
Main publications:
Varin V. P. An analysis of periodic flows in the vicinity of the plane Poiseille flow (with A. L.Afendikov) // Eur. J. Mech., B/Fluids, 10:6, 1991, 577–603.
Varin V. P. Bifurcations of some viscous fluid flow and transition to turbulence (with A. L. Afendikov) // Eur. J. Mech., B/Fluids, 10:2, Suppl., 1991, 13–18.
Varin V. P. The limit problems for the equation of oscillations of a satellite (with A. D. Bruno) // Celestial Mechanics and Dynamical Astronomy, 1997, v. 67, no. 1, pp. 1–40.
Varin V.P. Degeneracies of periodic solutions to the Beletskii equation // Regular & Chaotic Dynamics, 2000, v. 5, no. 3, pp. 38–54.
