This project report is to explore the behavior and stability of autonomous dynamical systems, focusing on their theoretical and practical applications. The primary objective of this study is to analyze the stability properties and to develop methods for assessing their robustness against perturbations.The project involves the fundamental concepts in dynamical systems theory, including equilibrium points, Jacobean matrix, Lyapunov’s direct method, linearization, and bifurcation analysis.The results indicate that the proposed methods provide effective tools for predicting systm behavior and ensuring stability and some real-life applications including electric circuts, damped pendules and Lotka-Volterra or predator-prey equations, were covered thoroughly