The Form of Key Function in the Problem of Branching of Periodic Extremals with Resonance 1:1:1

This article contains a method for calculating approximately the standardized key functions in the problem of branching of periodic extremals of a continuously differentiable action functional near its minimum. The periodic extremals of such functionals are used as a prototype for periodic oscillations of dynamical systems, ferroelectric crystal phases, nonlinear periodic waves, as so on. Recently Karpova, Kopytin, Derunova, and Sapronov studied cycle bifurcations in dynamical systems using key equations and key functions in the cases of double resonances $p_1:p_2:p_3$ with $p_1<p_2<p_3$. This article deals with the poorly understood case $p_1=p_2=p_3 = 1$. As a demonstration model, we consider an order six ODE. We use the Lyapunov–Schmidt method.