The book "Elements of Applied Bifurcation Theory" is a comprehensive text on bifurcation theory in dynamical systems, aimed at advanced undergraduate and graduate students, as well as researchers in physics, biology, engineering, and economics. It provides a solid foundation in dynamical systems theory and explains the approaches, methods, results, and terminology used in modern applied mathematics. The book covers major practical issues of applying bifurcation theory to finite-dimensional problems, including topological equivalence, codimension, and bifurcation diagrams. The second edition of the book updates the first edition with recent theoretical developments and improved numerical methods for bifurcation analysis. It includes major additions such as an elementary proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation, a detailed normal form analysis of the Neimark-Sacker bifurcation in the delayed logistic map, and explicit formulas for the critical normal form coefficients of all codim 1 bifurcations of n-dimensional iterated maps. The book also discusses homoclinic bifurcations, center manifolds, and bifurcations in symmetric systems. The text is structured into chapters that cover various aspects of bifurcation theory, including one-parameter and two-parameter bifurcations, homoclinic and heteroclinic orbits, and numerical analysis of bifurcations. Each chapter includes exercises and bibliographical notes, and the book provides explicit procedures for applying general mathematical theorems to particular research problems. It also emphasizes numerical implementation of the developed techniques and includes examples from mathematical biology. The book is written for a wide audience and is designed to be accessible, with a focus on practical applications and clear explanations. It avoids overly theoretical treatments and instead provides a balance between theory and application. The text is supported by numerous illustrations and examples, and it includes a detailed discussion of recent theoretical results, such as bifurcations of homoclinic orbits to nonhyperbolic equilibria and one-parameter bifurcations of limit cycles in systems with reflectional symmetry. The book also addresses important issues that are not covered in detail, such as chaotic dynamics and Hamiltonian systems, and provides introductory information on bifurcations in systems with symmetries.The book "Elements of Applied Bifurcation Theory" is a comprehensive text on bifurcation theory in dynamical systems, aimed at advanced undergraduate and graduate students, as well as researchers in physics, biology, engineering, and economics. It provides a solid foundation in dynamical systems theory and explains the approaches, methods, results, and terminology used in modern applied mathematics. The book covers major practical issues of applying bifurcation theory to finite-dimensional problems, including topological equivalence, codimension, and bifurcation diagrams. The second edition of the book updates the first edition with recent theoretical developments and improved numerical methods for bifurcation analysis. It includes major additions such as an elementary proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation, a detailed normal form analysis of the Neimark-Sacker bifurcation in the delayed logistic map, and explicit formulas for the critical normal form coefficients of all codim 1 bifurcations of n-dimensional iterated maps. The book also discusses homoclinic bifurcations, center manifolds, and bifurcations in symmetric systems. The text is structured into chapters that cover various aspects of bifurcation theory, including one-parameter and two-parameter bifurcations, homoclinic and heteroclinic orbits, and numerical analysis of bifurcations. Each chapter includes exercises and bibliographical notes, and the book provides explicit procedures for applying general mathematical theorems to particular research problems. It also emphasizes numerical implementation of the developed techniques and includes examples from mathematical biology. The book is written for a wide audience and is designed to be accessible, with a focus on practical applications and clear explanations. It avoids overly theoretical treatments and instead provides a balance between theory and application. The text is supported by numerous illustrations and examples, and it includes a detailed discussion of recent theoretical results, such as bifurcations of homoclinic orbits to nonhyperbolic equilibria and one-parameter bifurcations of limit cycles in systems with reflectional symmetry. The book also addresses important issues that are not covered in detail, such as chaotic dynamics and Hamiltonian systems, and provides introductory information on bifurcations in systems with symmetries.