This work is an empirical investigation of the nonlinear dynamics of stock markets with emphasis on Indian stock market. There are two main schools of time series analysis. The traditional one, which has a long pedigree in applied statistics, is prevalent among statisticians, social scientists (especially econometricians) and engineers. The more recent school, developed essentially since the 1970s, comes out of physics and nonlinear dynamics. The first views time series as samples from a stochastic process, and applies a mixture of traditional statistical tools and assumptions (linear regression, the properties of Gaussian distributions) and the analysis of the Fourier spectrum. The second school views time series as distorted or noisy measurements of an underlying dynamical system, which it aims to reconstruct. The authors use use methods from this second school to investigate the critical periods of stock market, as well as examine the underlying periodicity of different markets. In the end the celebrated Geometric Brownian Motion model of stock price movement is pitted against another model and the efficacy tested using these methods. The conclusions favour the newer model.