A feature parameter was proposed to quantitatively explore the boundedness of running-in attractors; its variation throughout the friction process was also investigated. The enclosing radius R was built with recurrence plots (RPs) and recurrence qualification analysis (RQA) by using the time delay embedding and phase space reconstruction. Additionally, the typology of RPs and the recurrence rate (RR) were investigated to verify the applicability of R in characterizing the friction process. Results showed that R is larger at the beginning, but exhibits a downward trend in the running-in friction process; R becomes smooth and trends to small steady values during the steady-state friction period, and finally shows an upward trend until failure occurs. The evolution of R, which corresponded with the typology of RPs and RR during friction process, can be used to quantitatively analyze the variation of the running-in attractors and friction state identifacation. Hence, R is a valid parameter, and the boundedness of running-in attractors can offer a new way for monitoring the friction state of tribological pairs.