A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

We deal with nonlinear TT-periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov–Schmidt reduction method applied to the TT-Poincaré–Andronov mapping.

► We study periodic solutions of a perturbed nonlinear TT-periodic differential system.
► Providing the second order bifurcation function in a small parameter whose simple zeros give the periodic solutions.
► The proof uses the Lyapunov–Schmidt reduction method applied to the TT-Poincaré map.
► Two applications are done.