Nonlinear dynamical systems underpin phenomena across fluid mechanics, epidemiology, climate science, and engineering, yet their inherent complexity renders classical analytical methods insufficient for real-world applications. This paper proposes and evaluates a Hybrid AI-Mathematical Modelling (HAIMM) framework that synergistically integrates physics-informed neural networks (PINNs), symbolic regression, and reservoir computing with traditional mathematical constructs such as ordinary and partial differential equations. The framework preserves the interpretability and physical consistency of mathematical models while harnessing the pattern-learning capacity of modern machine learning. Through four representative case studies Lorenz attractor forecasting, epidemic compartmental modelling, fluid boundary-layer prediction, and structural vibration identification we demonstrate that HAIMM achieves up to 94.3% prediction accuracy with a 40% reduction in computational overhead compared to purely numerical baselines. A structured comparison across model families is presented via tabular benchmarks, and key dynamical behaviors are illustrated through system-generated flow diagrams. The results suggest that hybrid approaches represent a principled, scalable pathway for taming the complexity of nonlinear dynamics in science and engineering.