Local dynamics for optimal control problems of three-dimensional ODE systems

This paper presents a complete characterization of the local dynamics for optimal control problems in three-dimensional systems of ordinary differential equations by using geometrical methods. The particular structure of the Jacobian implies that the sixth-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 three-dimensional space, which are homomorphic to the subspaces tangent to the complex, center and stable sub-manifolds. The main results are: a local stability theorem and necessary conditions for the existence of fold, Hopf, double-fold and fold–Hopf bifurcations.