In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

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