Dynamical instability induced by the initiation and advancement of mechanical faults in rotary elements is detrimental to the reliability and operation safety of the entire system. The inherent nonlinearity associated with bifurcation presents itself as difficulties in identifying and isolating features indicative of the presence and progression of faults that could lead to eventual mechanical deterioration. The perturbed and deteriorated states of a bearing-shaft system subjected to the actions of two types of commonly seen mechanical faults, namely, rotor speed and imbalance, are investigated using the basic notion of instantaneous frequency. The presented approach realizes temporal events of both short and long time scales as instantaneous frequencies in the joint time-frequency domain and thus effectively uncouples the harmonic components resulted from the coupling of multitude faults. Examples are given to demonstrate the feasibility of applying the approach to the characterization of various deteriorating bearing states and the identification of parameters associated with various modes of instability and chaotic response.