Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems

In this work, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of period-doubling bifurcations (PDB´s) starting from a small neighborhood of Hopf bifurcation points useful for analysis of multiple oscillations of periodic solutions for time-delayed feedback systems. The proposed algorithm employs higher order harmonic-balance approximations (HBA´s) for the predicted periodic solutions of the time-delayed systems. As compared to the same study of feedback systems without time delays, the HBA´s used in the new algorithm include only some simple modifications. Two examples are used to verify the graphical algorithm for prediction: one is the well-known time-delayed Chua´s circuit (TDCC) and the other is a time-delayed neural-network model