Nonlinear dynamics: A tutorial on the method of normal forms

Peter B. Kahn, Yair Zarmi

    Research output: Contribution to journalArticlepeer-review

    13 Scopus citations


    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.

    Original languageEnglish
    Pages (from-to)907-919
    Number of pages13
    JournalAmerican Journal of Physics
    Issue number10
    StatePublished - 1 Jan 2000

    ASJC Scopus subject areas

    • Physics and Astronomy (all)


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