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

Abstract

Abstract-In this brief, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of perioddoubling 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.

Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach

“…On the other hand, most time-domain methods for detecting bifurcations are based on the computation of the FM's since once the FM's are known, the bifurcation conditions are easily derived and the stability of the limit cycle is readily established as well. Unfortunately, the approaches proposed in [4]- [13] are not suitable for evaluating the FM's.…”

“…In order to have a nontrivial solution, the determinant of the infinite matrix [see (18) at the bottom of this page] must vanish where . It can be verified that equation has infinitely many solutions of the type ( and ) where are the eigenvalues defined in (13).…”

“…As was pointed out in Section I, several approaches have been proposed for studying stability and bifurcations of limit cycles detected through the HB technique [4]- [13]. On the other hand, most time-domain methods for detecting bifurcations are based on the computation of the FM's since once the FM's are known, the bifurcation conditions are easily derived and the stability of the limit cycle is readily established as well.…”

“…In [12] bifurcations are studied through a spectral technique based on the introduction of measuring probes into the circuit. In [13] bifurcations in time-delayed systems. However, none of the above approaches provides a method for computing the limitcycle Floquet multipliers (FM's), which are the simplest tool for establishing the stability of the limit cycle and for detecting its bifurcations.…”

Bifurcation analysis of a class of neural networks with delays

“…For n = 2, we refer to [2,4,[9][10][11][12]14,19,21,26,27,32,35], for n = 3, see [6,31,33,38,42], and for n = 4, see [20]. Another important models such as Cohen-Grossberg and BAM neural networks with delays have been studied extensively, we refer to [1,7,8,13,18,24,25,29,30] and references therein for stability analysis and existence of periodic solutions by constructing Liapunov functions.…”