Several pairs of algorithms were used to determine the phase space reconstruction parameters to analyze the dynamic characteristics of chaotic time series. The reconstructed phase trajectories were compared with the original phase trajectories of the Lorenz attractor, Rössler attractor, and Chen's attractor to obtain the optimum method for determining the phase space reconstruction parameters with high precision and efficiency. The research results show that the false nearest neighbor method and the complex autocorrelation method provided the best results. The saturated embedding dimension method based on the saturated correlation dimension method is proposed to calculate the time delay. Different time delays are obtained by changing the embedding dimension parameters of the complex autocorrelation method. The optimum time delay occurs at the point where the time delay is stable. The validity of the method is verified by combing the application of the correlation dimension, showing that the proposed method is suitable for the effective determination of the phase space reconstruction parameters.

In order to study the influence of rock combination types on their mechanical properties and failure characteristics, uniaxial compression tests of single rock samples and combined rock samples were conducted. Acoustic emission (AE) signals during the test process were collected, and the differences in AE signals of single rock samples and combined rock samples were studied based on the fractal theory. The results showed that the peak strength, elastic modulus, peak strain, and failure degree of the combined rock samples are all between those of the two single rock samples. The AE ringing count gradually increases with the loading process and suddenly increases to the maximum when the rock sample fails. During this process, the phase trajectory volume corresponding to the ringing count shows an evolution law of first decreasing and then increasing, while the correlation dimension corresponding to the ringing count signal shows an overall evolution law of first increasing and then decreasing. The results indicate that the phase trajectory volume, correlation dimension, and crack changes have a consistent dynamic change. Therefore, the phase trajectory and correlation dimension are effective tools to describe the pore change characteristics of rock combinations.

In this study, the coefficient of friction (COF) signals throughout the running-in process were examined by sliding a ring against a static disc. By reconstructing the scalar time-series into multi-dimensional phase spaces, friction-induced attractors were obtained and quantified by recursive characteristics analysis, which can effectively realize the running-in status identification. Moreover, a recursive characteristics analysis-based evaluation model was established to investigate the stationarity of the friction-induced attractors based on the recurrence quantification analysis (RQA) measures. The analyses of the numerically simulated signals and experimental results indicate that the extracted model is an intuitive and effective method. Furthermore, to improve the stationary of the friction-induced attractors, the normal pressure should be low, whereas the relative sliding velocities should be increased appropriately. These results would contribute to the revelation of the recursive characteristics of the tribosystem and the improvement of the stationarity of friction-induced attractors.

A Monte Carlo based fractal prediction model is proposed to describe the evolution of surface morphology during the running-in process. The model accounts for the random and fractal characteristics of worn surfaces. The Weierstrass–Mandelbrot function is employed to simulate rough surfaces and establish the correlation between fractal dimension and surface roughness. By integrating traditional sliding wear models with surface effect functions, a unified prediction framework is developed. Experiments are conducted to obtain worn surface parameters and calculate fractal dimensions at different running-in stages. Model parameters are optimized by minimizing the variance between experimental and predicted results. Monte Carlo simulations are then introduced to represent the stochastic nature of the friction system, thereby improving prediction accuracy and objectivity. The proposed model reveals locally random yet globally convergent patterns, which are consistent with experimental observations. It effectively captures the stochastic evolution of surface morphology and provides a reliable approach for predicting worn surface behavior during running-in.