Nonlinear Dynamics

Normal forms are among the most powerful mathematical tools available to researchers investigating nonlinear dynamical systems. Using the normal forms method, physicists and engineers can simplify complex systems in order to isolate and study, with relative ease, the vibrations, oscillations, bifurcations, and other dynamical attributes of those systems. In the first book devoted exclusively to exploring the many possibilities afforded by this important investigative tool, Professors...
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Normal forms are among the most powerful mathematical tools available to researchers investigating nonlinear dynamical systems. Using the normal forms method, physicists and engineers can simplify complex systems in order to isolate and study, with relative ease, the vibrations, oscillations, bifurcations, and other dynamical attributes of those systems. In the first book devoted exclusively to exploring the many possibilities afforded by this important investigative tool, Professors Peter B. Kahn and Yair Zarmi develop a detailed exposition of normal forms as applied to nonlinear systems modeled by differential equations that are amenable to perturbative analysis. Throughout the book the authors emphasize the freedom or nonuniqueness inherent to the normal form expansion and go to lengths to demonstrate clearly how it enables researchers to obtain perturbative expansions that are numerically far superior to those obtained by approaches that ignore inherent freedom. Nonlinear Dynamics begins with an introduction to the basic concepts underlying the normal forms method and the role of freedom in the near-identity transformation that is the key to its development. Coverage then shifts to an investigation of systems with one degree of freedom (con-servative and dissipative) that model electrical and mechanical oscillations and vibrations where the force has a dominant linear term and a small nonlinear one. The authors consider the rich variety of nonautonomous problems that arise during the study of forced oscillatory motion. Topics covered include boundary value problems, connections to the method of the center manifold, linear and nonlinear Mathieu equations, pendula, orbits in celestial mechanics, electrical circuits, nuclear magnetic resonance, and resonant oscillations of charged particles due to multipole errors in guiding magnetic fields in particle accelerators. Providing the most detailed coverage of the subject currently available and featuring numerous examples, Nonlinear Dynamics serves equally well as a professional reference for engineers, and a course text for advanced-level students in nonlinear dynamics in physics, applied mechanics, and applied mathematics. Nonlinear Dynamics uses the normal forms method to explore one of the most fertile research areas today. This text, the first exclusively devoted to an extensive exposition of the normal forms method, affords physicists and engineers a unique opportunity to explore the enormous potential of this powerful investigative approach to describing the behavior of a wide range of complex phenomena. Provides the most detailed coverage of the normal forms method available Emphasizes the nonuniqueness property of normal form expansion throughout Features numerous examples drawn from electrical and mechanical engineering, particle physics, astrophysics, and other scientific disciplines