We study a control law, based on the extremum seeking method, which allows a nonholonomic unicycle to find the source of a scalar signal and, once found, remain in its vicinity.  The vehicle, with constant forward velocity and tuned angular velocity, does not have the capability of sensing its own position or the position of the source but is capable of sensing the scalar signal which originates from the source and decays away from it. Extremum seeking provides "persistency of excitation" to generate a gradient estimate in a non-model based fashion and is equipped to handle the kinematic constraints of the vehicle.  However, because of the constant forward velocity constraint, after the vehicle has converged near the source, it exhibits interesting but very complex motions as it repeatedly passes over the source, turns around, and heads back towards the source, or "hovers" around the source, never settling, and remaining alert to possible changes in the location of the source.

We identify these almost periodic attractors (combining the frequency of probing in extremum seeking with the, possibly incommensurate, frequency of hovering around the source) and use the averaging theory to prove the attractors' local exponential stability. We also study the situations where the vehicle heads towards the source perfectly and no unbiased strategy exists for deciding whether to turn back clockwise or counterclockwise after passing over the source. We characterize the effect of this topological obstacle in the problem by identifying a measure zero set of solutions that may theoretically take the vehicle to infinity, although, in practice, any small amount of noise will result in the vehicle diverging away from such unstable solutions. We analyze both 2D and 3D applications.