A new method of nonlinear analysis is established by combining phase space reconstruction and data reduction sub-frequency band wavelet. This method is applied to two types of chaotic dynamic systems (Lorenz and Rössler) to examine the anti-noise ability for complex systems. Results show that the nonlinear dynamic system analysis method resists noise and reveals the internal dynamics of a weak signal from noise pollution. On this basis

the vertical upward gas–liquid two-phase flow in a 2 mm× 0.81mmsmall rectangular channel is investigated. The frequency and energy distributions of the main oscillation mode are revealed by analyzing the time– frequency spectra of the pressure signals of different flow patterns. The positive power spectral density of singular-value frequency entropy and the damping ratio are extracted to characterize the evolution of flow patterns and achieve accurate recognition of different vertical upward gas–liquid flow patterns (bubbly flow: 100%

slug flow: 92%

churn flow: 96%

annular flow: 100%). The proposed analysis method will enrich the dynamics theory of multi-phase flow in small channel.