| The paper presents a complete characterization of the local dynamics for optimal control problems of 4-dimensional systems of ordinary differential equations, by using geometrical methods. We prove that the particular structure of the Jacobian implies that the 8 th order characteristic polynomial is equivalent to a composition of two lower order polynomials, which are solvable by radicals. The classification problem for local dynamics is addressed by finding partitions, over an intermediate 4-dimensional space, which are homomorphic to the sub-spaces tangent to the complex, center and stable sub-manifolds. Then we get local necessary conditions for the existence of 1- to 4-fold, Hopf, 1- and 2-fold-Hopf and Hopf-Hopf bifurcations, and represent them geometrically. |