On the stability of smooth branches of periodic solutions for higher order perturbed differential equations

The averaging method combined with the Lyapunov-Schmidt reduction provides sufficient conditions for the existence of periodic solutions of the following class of perturbative $T$-periodic nonautonomous differential equations $x^{\mathrm{\prime }}=F_0(t,x)+\epsilon F(t,x,\epsilon )$. Such periodic solutions bifurcate from a manifold $\mathcal{Z}$ of periodic solutions of the unperturbed system $x^{\mathrm{\prime }}=F_0(t,x)$. Determining the stability of this kind of periodic solutions involves the computation of eigenvalues of matrix-valued functions $M(\epsilon )$, which can done using the theory of $k$-hyperbolic matrices. Usually, in this theory, a diagonalizing process of $k$-jets of $M(\epsilon )$ must be employed and no general algorithm exists for doing that. In this paper, we develop an alternative strategy for determining the stability of the periodic solutions without the need of such a diagonalization process, which can work even when the diagonalization is not possible. Applications of our result for two families of $4$D vector fields are also presented.