This paper presents a comprehensive review and extension of the author’s work over the past three decades on two key power system phenomena, namely voltage collapse and sub-synchronous resonance (SSR), using advanced nonlinear analysis methods such as bifurcation and chaos theory. Voltage collapse is treated as a dynamic instability that arises when the grid can no longer sustain acceptable voltage levels under increasing load or reactive power demand. In contrast, SSR is described as an interaction between the electrical network and the turbine, generator shaft at frequencies below the nominal synchronous frequency, which can induce harmful torsional vibrations and potential mechanical damage. In the study of voltage collapse, nonlinear methods were applied to a three-bus system by developing a model comprising four nonlinear differential equations, with load power ($P_L$ or $Q_L$) serving as the control parameter. The findings indicate that as these load parameters increase, the inherent nonlinear characteristics of the system trigger a gradual loss of stability. A comparable modeling strategy was used to analyze SSR. A real-world system, the Mohave Power Station in Nevada, is examined, in which power is transmitted over a long line equipped with a series capacitor ($X_C$) to minimize AC losses ($I^2 X_L$). Despite these measures, catastrophic failures occurred at the station in 1970 and 1971 owing to turbine, generator shaft fractures, resulting in total system breakdown. For SSR analysis, a nonlinear model with 28 differential equations was created, using $X_C$ as the control parameter. The results show that increasing $X_C$ pushes the system toward instability, and surpassing a certain threshold leads to system collapse, aligning with documented shaft failures. Overall, the research highlights how modern nonlinear theories, such as bifurcation and chaos analysis, are essential for uncovering and anticipating complex dynamic behaviors in power systems.