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The book aims to provide a student or researcher with a basis in the dynamical systems theory and to give them the necessary understanding of the approaches, methods, results and terminology used in the modern applied mathematics literature. The book covers the basic topics of the bifurcation theory and can help to compose a course on nonlinear dynamical systems or system theory. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for PhD students and researchers in physics, biology, engineering and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used.