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

Abstract

In this brief, 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. I. INTRODUCTION In the last two decades, there has been continuously increasing interest in studying the coexistence of periodic and chaotic behaviors in nonlinear dynamical systems, such as some electronic devices and physiological organisms. Interdisciplinary research among specialists with different backgrounds from applied and computational math- ematics, circuits and systems engineering, physical and biomedical sciences, etc., has been highly motivated and, recently, has become very active. In particular, research on analysis of dynamical behaviors of a living or artificial neural network (modeled by nonlinear circuits) has attracted much attention from the scientific and engineering communities. In a (living or artificial) neural network, the special feature of multi- ple oscillations is a mechanism that has been recognized as important for memory storage. Therefore, detecting the coexistence of periodic and chaotic behaviors in a general nonlinear dynamical system has a great impact on understanding and utilizing complex networks; for