During the study the dynamics of controlled movement of tether systems, it is proposed to use topological structures of the exponential law of the length control of the connecting tether. Using the example of a flat circular motion of the mass center of the tether system proved the application of the exponential law, which allows us to obtain a finite number of topological structures. At each time point the control parameter of the exponential control law and, consequently, the corresponding topological structure can be determined for any control law of the length of the connecting tether. The comparison of the control law under consideration and the range of changes in the control parameter of the exponential law makes it possible to identify all the “slippery” parts of the relative motion of connected objects in a “kaleidoscope” of topological structures and explain the source of stochastic movements.