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On a Global Sequential Scenario of Bifurcation Trees to Chaos in a First-Order, Periodically Excited, Time-Delayed System

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.