Orbital stability of a small periodic solution of an autonomous system of differential equations

We consider a normal autonomous system of differential equations with a small parameter, which has a critical linear approximation at zero value of the parameter. We introduce the concept of orbital stability with respect to the parameter; according to this concept, the closeness of the right semitrajectories is achieved not only due to the proximity of initial values of solutions, but also due to the smallness of the parameter. We examine the problem of branching of a stable periodic solution with a period close to the period of solutions of the corresponding linear homogeneous system. Sufficient conditions for the solvability of the problem are established. Our reasonings are based on the properties of the first homogeneous nonlinear approximation of the monodromy operator.