ADVANCES IN NONLINEAR MATHEMATICAL MODELS FOR COMPLEX PROBLEM SOLVING EXPLORING INNOVATIVE APPROACHES TO REPRESENT, SIMULATE, AND OPTIMIZE NONLINEAR SYSTEMS

The mathematical models which are non-linear have become the pivot to the process of cognition, description and resolution of a vast array of complex problems that are relevant to irrelevant areas of life as many as engineering, physics, biology, economics, and social dynamics. The system behavior of wide bifurcations, chaos, and nonlinear initial condition behavior like behavior is characteristic of nonlinear systems and requires more complicated analytical, computational and optimization algorithms. This paper provides an overview of the current trends in nonlinear mathematical modeling with the focus on more recent methods of representation, simulation, and optimization of complex systems. These methodologies include the nonlinear differential equations, agent-based models, fuzzy logic, neural networks, and hybrid optimization models, which are highlighted. The research paper discusses computational methods, including numerical methods, metaheuristic as well as high-performance simulation methods as solutions to the nonlinearity menace. These superior models have been demonstrated to introduce a high threshold of predictive accuracy, systems endurance as well as decision making attainments. The complexity of the computation, however, limits its practical implementation due to the sensitivity of parameter estimation and the problem of validation of the model. Scenarios here the directions in the future will suggest a combination of machine learning and nonlinear modeling, designing adaptive and real-time simulation, and field crafting scalable algorithms of large-scale nonlinear systems.