Oscillations may cause both economic and technical problems such as a reduction in overall system reliability. Therefore, detecting and preventing oscillatory behavior that affects power systems is important. This paper proposes an oscillation recognition method that includes monitoring and extracting features in a recursive and sequential manner in a time-series measurement in power systems. We propose a geometric feature extraction process for recognizing oscillations by constructing an average system and Poincaré map for time-series measurement. The proposed process provides the features of a system's damping and frequency of oscillation, and the developed monitoring systems are based on nonlinear dynamics. The circulating oscillatory behavior is represented on a finite-integer-delay embedded time-series plane, extracted by a Poincaré map construction, and examined directly along the trajectory to monitor the features of the oscillation according to damping and frequency. Oscillatory behavior recognition is tested on IEEE's second benchmark system for subsynchronous resonance to verify the fast extraction of oscillation components. In addition, a case study for Korean power systems with a high penetration of renewable energy and application on actual measurement data is carried out to demonstrate the practical application of the process.