Dynamical and vibratory systems are an application of mathematics and applied sciences to the solution of real-world problems. Before being able to solve real world problems, it is necessary to carefully study dynamical and vibratory systems and solve all available problems in case of linear and nonlinear equations using analytical and numerical methods. Almost all applied processes act nonlinearly, and nonlinear analysis of complex systems is one of the most important and complicated tasks, especially in engineering and applied sciences problems. Other related books are written at a fundamental level that may not meet ambitious engineering program requirements or are specialized in certain fields of oscillatory systems, including modeling and simulations. This book presents a balance between theory and practice, fundamentals and advanced subjects, and generality and specialization. None of the books have completely studied and analyzed nonlinear equation in dynamical and vibratory systems using the latest analytical and numerical methods. Thereby in this book, by the use of the latest analytic, numeric laboratorial methods and using more than 300 references-like books, papers and the researches done by the authors and by considering almost all possible processes and situation, new theories have been proposed to encounter applied problems in engineering and applied sciences, allowing the user (bachelor’s, master’s and PhD students, university teachers and others in fields of mechanical, civil, aerospace, electrical, chemical, applied mathematics, physics, and etc.) can encounter such systems confidently. In the book, applied examples are practically solved by the proposed methodology.