In the modern precision machinery industry, although gas bearings (GBs) are commonly used in ultraprecision machining, precision measurement, and high-speed cutting, the irregular motion of a GB, i.e., nonlinear behavior, is often found under high-speed rotation owing to rotor eccentricity and gas characteristics. Herringbone-grooved journal gas bearings (HJGBs) have been increasingly used in mechanisms requiring precision rotation owing to their high stability and the better support force applied to the rotor than traditional GBs. However, the strong nonlinearity of the function describing the air film pressure and the dynamic problems of actual bearing systems, including inappropriate rotational speed, unbalanced gas supply, and improper bearing design, will cause the nonperiodic or chaotic motion and instability of the rotor-bearing system under some conditions. Such irregular motion will damage machines in some situations. To determine the conditions that cause the rotor of HJGBs to undergo nonperiodic motion, and thereby prevent irregular vibration, we apply embedded piezo vibration sensors in an HJGB system to investigate in detail the relevant bearing characteristics and verify the results. To examine the nonlinear behavior of rotors, Poincaré maps and the maximum Lyapunov exponent are used to detect the vibration signal from piezo sensors, and GoogLeNet and ResNet-50 are used to identify the type of vibration behavior. The results of rotor behavior diagnosis show that the accuracies of GoogLeNet and ResNet-50 were 100 and 99.7%, respectively, i.e., GoogLeNet outperformed ResNet-50. This paper provides a diagnosis of the occurrence of chaos in an HJGB and a means of inhibiting nonperiodic motion to reduce the system loss caused by irregular vibration.