Chaos is a fundamental property that comes out of nonlinearity and a sensitive dependence on initial condition. A nonlinear dynamic system that displays a chaotic steady state behaviour is purely deterministic, but its long-term behaviour cannot be predicted because of the property of sensitivity to initial conditions. One of the most promising applications of chaos theory, which exploits both the deterministic and unpredictable aspects of chaotic behaviour, is chaos-based secure communications. The sensitivity to initial condition is quantified with the help of the entropy named Lyapunov Exponent. Generation of chaos from nonlinear optical devices have been explored from several years. In this paper an EO modulator system of cos2 type of nonlinearity is selected as chaos producing device and characterized with the help of Lyapunov Exponent. The Lyapunov Exponent is computed here to determine the dependence of a chaotic signal on the initial conditions and also on the system parameters. The period doubling behaviour in this Electro optic modulator system have also been studied here. The transition from periodic to chaotic behaviours in this one-dimensional discrete dynamical system have been controlled through variation of a single parameter out of different system parameters. With the help of Lyapunov Exponent and bifurcation analysis technique an estimation of the system parameters has been carried out to predict the nature of the output generated at the terminal of the EO cell.