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. 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 (conservative 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.