Family of Logarithmic Spirals in Hamiltonian Systemsof Dimension 8 with Control in a Disk

We study a neighbourhood of a singular second-order extremal in optimal control problems that are affine in control in a disk. We consider the case in which the Hamiltonian system has dimension 8 and is a small (in the sense of the action of the Fuller group) perturbation of the Hamiltonian system of the generalized Fuller problem with control in the disk. For this class of problems, we prove the existence of extremals in the form of logarithmic spirals that reach the singular second-order extremal in a finite time, while the control performs infinitely many rotations around the circle.