Structural stability of logarithmic spirals in control problems with second-order singular extremal

A nonlinear perturbation of generalization of the Fuller problem with controls in a disk is considered. The structural stability of logarithmic spirals is studied. It was shown that if perturbations are small with respect to the action of the symmetry group of the unperturbed problem, then in the neighborhood of a singular second-order solution, extremals in the form of logarithmic spirals are preserved. The constructed extremals arrive at a singular extremal in a finite time, while the controls make an infinite number of revolutions along the circle.