We consider oscillations observed in the mathematical model describing the dynamics of transmembrane potential and changes of proton concentration outside and inside the cell of alga Chara corallina. This model shows sufficiently nonlinear regimes such as periods of birth/disappearance as well as parameter regions of chaotic motion. We perform bifurcation analysis of these complex oscillations via the Continuous Wavelet Transform with the Morlet wavelet. We show that this method, compared to classical bifurcation analysis allows to make numerical simulations at the continuous change of the key parameters and to obtain the spectral characteristics.