Bifurcation of Periodic Solutions for $C^5$ and $C^6$ Vector Fields in $R^4$ with Pure Imaginary Eigenvalues in Resonance 1:4 and 1:5

In this paper we study the bifurcation of families of periodic orbits at a singular point of a C^5 and C^6 differential system in R^4 with pure imaginary eigenvalues with resonance 1:4 and 1:5 respectively. From the singular point of the C^5 vector field with resonance 1:4 can bifurcate 0, 1, 2, 3, 4, 5 or 6 one-parameter families of periodic orbits. For the C^6 vector field with resonance 1:5, the maximal number of families of periodic orbits that bifurcate from this singular point is $40$. The tool for proving such a result is the averaging theory.