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    DYNAMICAL SYSTEMS 1 First semester : For first-year Master’s degree students
    (Faculty of Sciences, 2025) Boulfoul , Amel
    This course book serves as an introduction to dynamical systems, focusing on ordinary differential equations (ODEs), their stability, periodic solutions, and bifurcations. It is aimed at students in mathematics, particularly those in the first year of their Master’s degree. The book is structured into five main chapters, each addressing important aspects of dynamical systems. The first chapter, Preliminaries of Ordinary Differential Equations, begins with a review of differential systems, followed by their classification, and linear differential systems, including homogeneous and nonhomogeneous cases. The second chapter, General Theory of ODEs, introduces the fundamental aspects of ordinary differential equations, initial value problems, and solutions. It includes key existence and uniqueness theorems, different proof methods, and examples. Additionally, the continuation of solutions and maximal intervals of existence are discussed in detail. The third chapter, Stability in Linear and Nonlinear Systems, explores the concept of stability, starting with linear systems and extending to nonlinear systems. It covers important tools such as Lyapunov’s method and the analysis of conservative and dissipative systems, which are fundamental for understanding system behavior over time. In the fourth chapter, Periodic Solutions and Their Stability, we delve into the nature of periodic solutions, limit cycles, and their stability. Concepts such as Poincar´e maps, Bendixson’s and Dulac’s criteria, and the Poincar´e-Bendixson theorem are explored, offering a deep understanding of the behavior of dynamical systems in the long term. Finally, the fifth chapter, Introduction to Local Bifurcations, introduces the v concept of bifurcation, focusing on one-dimensional and two-dimensional systems. Key bifurcations, including saddle-node, pitchfork, transcritical, and Hopf bifurcations, are explored, laying the foundation for further study and applications in complex systems. In this book, my objective was to collect the most important definitions, information, and tools from the most significant references such as [[12], [14], [15], [10], [5]], and the references therein, to help students focus on the essential notions of dynamical systems. Throughout the book, a formal and clear approach is taken, with numerous examples and illustrations. The topics covered are foundational for the study of dynamical systems and provide the necessary tools to tackle more advanced subjects in mathematical modeling, control theory, and applied mathematics. It is my hope that this book serves as a learning resource for students and a reference for those seeking to deepen their understanding of dynamical systems