Chaos is a property of dynamical system which is completely unordered, unpredictable, uncontrolled and has a sensitive dependence on initial condition as the main features of chaos. This book presents some analyses on chaotic dynamical systems. We study on the findings of the critical value’s formula for the chaotic system by using the linearized system and characteristic equations for chaotic dynamical flows. Some numerical methods are used to solve the chaotic systems for maps and flows. We also investigate about the sensitivity to initial conditions which is measured by the Lyapunov exponents for both maps and flow systems. Lyapunov exponent measures the rate of divergence or convergence of two nearby initial points and is also used to detect the presence of chaos behavior. In the last chapter of this book, we propose a new chaotic system that we discovered recently by modifying the Zhou’s system (2008). We examine some of its basic dynamical properties and this new system is proven to be chaotic within a certain range of parameters.