Foundations of Mathematical Modeling and Dynamical Systems with Applications
Dr. K. Ramesh *
Anurag University, Hyderabad, Telangana, India.
Dr. G. Ranjith Kumar
Anurag University, Hyderabad, Telangana, India.
Dr. K. Lakshmi Narayan
Department of Humanities & Sciences, VJIT, Hyderabad, Telangana, India.
Dr. A. V. Papa Rao
Department of Mathematics, JNTU-GV College of Engineering, Vizianagaram, Andhra Pradesh, India.
*Author to whom correspondence should be addressed.
Abstract
One of the most effective methods for comprehending, evaluating, and forecasting the behaviour of real-world systems that arise in the fields of science, engineering, biology, economics, and the social sciences is mathematical modelling. Models give researchers a methodical framework for investigating system dynamics, testing theories, and directing decision-making by converting physical, biological, or socioeconomic processes into mathematical language.
The goal of this book is to present a thorough, organised, and understandable introduction to mathematical modelling based on continuous dynamical systems, with a focus on ordinary differential equations. The presentation integrates theoretical foundations, qualitative analysis, and computational tools, allowing readers to transition seamlessly from model creation to analytical insight and numerical exploration.
The introductory chapters present essential ideas of mathematical modelling, encompassing modelling assumptions, variable selection, parameter interpretation, and dimensional consistency. Continuous first-order differential equations are formulated using compelling examples from natural and applied sciences, thus building a robust conceptual foundation.
The qualitative theory of dynamical systems, including linearization, equilibrium analysis, and stability theory, is the main topic of the following chapters. These methods make it possible to comprehend the behaviour of long-term systems without depending on explicit solutions, which are frequently not available for nonlinear models.
Then, bifurcation theory is introduced to show how qualitative changes in system dynamics can result from minor changes in parameters. In applied models, bifurcations that are often encountered are given particular consideration. Biological, ecological, and engineering systems are used to illustrate the analytical and geometric methods for identifying and characterizing periodic solutions that are developed in the limit cycles chapter.
The final chapter is dedicated to simulation techniques, acknowledging the essential role of computation in contemporary modelling. Numerical approaches, phase-plane simulations, and computational tools like MATLAB are employed to enhance analytical findings and to explore intricate systems that exceed closed-form analysis.
This book is designed for advanced undergraduate and graduate students, along with academics and practitioners pursuing a comprehensive introduction to mathematical modelling and dynamical systems. The information is appropriate for courses in applied mathematics, mathematical biology, engineering mathematics, and associated fields. The literature reinforces theory using examples, pictures, and simulations to enhance intuition and practical comprehension.
It is intended that this book will give readers the mathematical rigour and modelling understanding they need to examine real-world occurrences and create insightful models for a variety of application domains
Keywords: Mathematical modelling, dynamical systems, ordinary differential equations (ODEs), stability and bifurcation theory