Mathematical modelling has emerged as one of the most powerful tools for understanding, analysing, and predicting the behaviour of real-world systems arising in science, engineering, biology, economics, and social sciences. By translating physical, biological, or socio-economic processes into mathematical language, models provide a systematic framework for exploring system dynamics, testing hypotheses, and guiding decision-making.
This book aims to provide a clear, structured, and rigorous introduction to mathematical modelling based on continuous dynamical systems, with particular emphasis on ordinary differential equations. The presentation balances theoretical foundations, qualitative analysis, and computational techniques, enabling readers to progress naturally from model formulation to analytical insight and numerical exploration.
The initial chapters introduce fundamental principles of mathematical modelling, including modelling assumptions, variable selection, parameter interpretation, and dimensional consistency. Continuous first-order differential equations are developed through motivating examples drawn from natural and applied sciences, establishing a strong conceptual foundation.
Subsequent chapters focus on the qualitative theory of dynamical systems, including linearisation, equilibrium analysis, and stability theory. These tools allow long-term system behaviour to be understood without reliance on explicit solutions, which are often unavailable for nonlinear models.
Bifurcation theory is then presented to demonstrate how small parameter variations can lead to significant qualitative changes in system dynamics. Special attention is given to bifurcations commonly encountered in applied models. The chapter on limit cycles develops both analytical and geometric approaches for identifying and characterising periodic solutions, with applications drawn from biological, ecological, and engineering systems.
Recognising the essential role of computation in modern modelling, the final chapter is devoted to simulation techniques. Numerical methods, phase-plane analysis, and computational tools such as MATLAB are introduced to complement theoretical results and to investigate complex systems beyond closed-form solutions.
This book is intended for advanced undergraduate and postgraduate students, research scholars, and teachers in applied mathematics, mathematical biology, engineering mathematics, and related disciplines. The material is presented in a clear and systematic manner, supported by examples, figures, and simulations to enhance conceptual understanding and practical skills.
