The paper further developed the methodological approach in obtaining mathematical models that describe the running-in of tribosystems under boundary lubrication conditions.The structural and parametric identification of the tribosystem as an object of simulation of running-in under conditions of extreme lubrication was carried out. It has been established that the processes of running-in of tribosystems are described by a second-order differential equation and, unlike the known ones, take into account the limit of loss of stability (robustness reserve) of tribosystems. It is shown that the nature of tribosystems running-in conditions of extreme lubrication depends on the gain coefficients and time constants, which are included in the right-hand side of the differential equation. It is shown that the processes of running-in of tribosystems depend on the type of the magnitude of the input influence on the tribosystem, the first and second derivatives. The input influence is represented as a product of coefficients and a time constant К0·К2·Т3. This allows us to state that the processes of the tribosystem running-in will effectively take place when the input action (load and sliding speed), will change in time and have fluctuations with positive and negative acceleration of these values from the set (program) value. This requirement corresponds to the running-in program "at the border of seizing".The left part of the equation is the response of the tribosystem to the input signal. Tribosystem time constants Т2 and Т3  have the dimension of time and characterize the inertia of the processes occurring in the tribosystem during running-in. Increasing the time constants makes the process less sensitive to changes in the input signal, the warm-up process increases in time, and the tribosystem becomes insensitive to small changes in load and sliding speed. Conversely, the reduction of time constants makes the tribosystem sensitive to any external changes