Simple equations exist relating adhesion and friction forces. These apply to simple attachment-detachment processes and steady-state ( smooth) sliding conditions. However, in the case of more complex, such as polymer, surfaces both the adhesion and friction can be very complex, irreversible, and nonlinear, exhibiting stringing and tack in the former and stick-slip sliding in the latter. We explore possible relationships between such nonlinear adhesion and friction processes. Based on recent experiments we find that certain types of "limit cycles,'' relating the ( normal) adhesion and (lateral) friction forces, F(perpendicular to) and F(parallel to), to the relative velocities, V(perpendicular to) and V(parallel to), of the surfaces during an attachment-detachment process or stick-slip sliding, bear a very similar resemblance to each other. We briefly discuss the theoretical and practical implications of describing such dynamic processes in terms of limit cycles.