Application of normalized key functions in a problem of branching of periodic extremals

In this paper we construct a procedure of approximate calculation and analysis of branches of bifurcating solutions to a periodic variational problem. The goal of the work is a study of bifurcation of cycles in dynamic systems in cases of double resonances $1:2:3$, $1:2:4$, $p:q:p+q$ and others. An ordinary differential equation (ODE) of the sixth order is considered as a general model equation. Application of the Lyapunov–Schmidt method and transition to boundary and angular singularities allow to simplify a description of branches of extremals and caustics. Also we list systems of generators of algebraic invariants under an orthogonal semi-free action of the circle on $\mathbb R^6$ and normal forms of the main part of the key functions.