Non-linear dynamics is a branch of mathematics and physics that studies systems where changes in input do not produce proportional changes in output. This means that small differences in initial conditions can lead to vastly different outcomes, a phenomenon often referred to as the butterfly effect. Non-linear systems are common in nature, appearing in weather patterns, population dynamics, and even in the behavior of financial markets. In contrast to linear systems, which can be easily predicted and analyzed, non-linear systems can exhibit complex behaviors such as chaos, bifurcations, and oscillations. Understanding non-linear dynamics is crucial for fields like engineering, biology, and economics, as it helps scientists and researchers model and predict the behavior of intricate systems that are influenced by multiple interacting factors.