We consider a variety of nonlinear systems, described by linear differential equations, subjected to small nonlinear perturbations. Approximate solutions are sought in terms of expansions in a small parameter. The method of normal forms is developed and shown to be capable of constructing a series expansion in which the individual terms in the series correctly incorporate the essential aspects of the full solution. After an extensive introduction, we discuss a series of examples. Most of our attention is given to autonomous systems with imaginary eigenvalues for the unperturbed problem. But, we also analyze a system of equations with negative eigenvalues; one zero and one negative eigenvalue; two nonautonomous problems and phase locking in a coupled-oscillator system. We conclude with a brief section on an integral formulation of the method.
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