The study of one parameter families of differential equations is extremely important in order to determine and characterize their harmonic solutions. The main purpose of this paper is to prove, under the assumptions that the Hopf index of ``\(w\)'' (\(w: M\to\mathbb{R}^k\) is the average wind velocity associated with the map \(f\)) in \(\Omega\cap M\) is well defined and non-zero, the existence of a connected subset \(\Gamma\) of \(\Omega\) (\(\Omega=\) an arbitrary open subset of the Cartesian product \([0,\infty)\times C_T(M)\); \(C_T(M)=\) the metric subspace of \(C_T(\mathbb{R}^k)\) consisting of those maps whose images lie in \(M\); \(M=\) a differentiable variety of the following one parameter differential equations: \(\dot x=\lambda f(t,x)\), \(\lambda\geq 0\); \(f:\mathbb{R}\times M\to\mathbb{R}^k\) is a \(T\)-periodic vector field, tangent to a boundaryless smooth submanifold of \(\mathbb{R}^k\); \(C_T(\mathbb{R}^k)=\) the Banach space of all continuous, \(T\)-periodic, \(\mathbb{R}^k\)-valued real maps, endowed with the standard norm of uniform convergence), satisfying certain conditions. More exactly, the authors develop and demonstrate a method in order to deduce the existence of the above global bifurcation branch \(\Gamma\) of nontrivial solution pairs from the finite-dimensional results of their previous papers.