An iterative method is proposed for finding periodic orbits of strongly nonlinear oscillators. The method combines the strength of analytical approaches, where the candidate solution is assumed in the form of a Fourier series, and the convenience of numerical methods that can be applied to larger systems with strong nonlinearity. The proposed method does not require integration of the vector field over any period of time and examples presented here illustrate that it is faster than traditional collocation algorithms, has a large radius of convergence, and is capable of finding several periodic orbits in each solution.

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