An improved version of a frequency-domain approach to study bifurcations in delay-differential equations is presented. The proposed methodology provides information about the frequency, amplitude, and stability of the orbit emerging from Hopf bifurcation. We apply this method to different schemes of the delayed van der Pol oscillator. The time-delay dependence can appear intrinsically because of the system dynamics or can be intentionally introduced in a feedback loop. Also, a discussion about system controllability and observability is given for a proper and rigorous application of the frequency domain technique. Collateral findings involving some types of static bifurcations are included for completeness.
The effect of delayed feedback on the dynamics of a scalar map is studied by using a frequency-domain approach. Explicit conditions for the occurrence of period-doubling and Neimark-Sacker bifurcations in the controlled map are found analytically. The appearance of a 1:2 resonance for certain values of the delay is also formalized, revealing that this phenomenon is independent of the system parameters. A detailed study of the well-known logistic map under delayed feedback is included for illustration.
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