Bifurcations and Hopf Degeneracies in Nonlinear Feedback Systems With Time Delay

Abstract

An application of the well-developed frequency-domain approach to detect oscillations in nonlinear feedback systems with time delay is presented. The method depends on an early proof of the Hopf bifurcation theorem known as the Graphical Hopf Theorem (GHT). Several nondegeneracy conditions are included to apply the GHT in nonlinear systems with time delay. The singular conditions corresponding to degeneracies, which include static and dynamic bifurcations, as well as some special cases of degenerate Hopf bifurcations and multiple crossings, are also discussed. Two Single-Input Single-Output (SISO) feedback systems with odd nonlinearities are presented as examples to show that the proposed technique and a standard simulation method have very good agreement in the results, yet the GHT is much simpler in calculation. The first one shows an application of the GHT under classical Hopf conditions while the second emphasizes the presence of degenerate Hopf bifurcations and multiple crossings. For both examples, and others which have appeared recently in the literature, a considerable simplification of the formulas for recovering periodic solutions is also provided in this paper.

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

“…Input delay is resulted mainly from the transmission of control signals, or is introduced deliberately for control purposes [12]. Systems with input delays are studied by many researchers [13][14][15][16][17][18][19][20][21][22][23].…”

RESONANCES OF A NON-LINEAR s.d.o.f. SYSTEM WITH TWO TIME-DELAYS IN LINEAR FEEDBACK CONTROL

“…They gave the results in a number of "gures for the steady state response amplitude versus the excitation frequency and amplitude. Moiola et al [5] investigated the degeneracy conditions of Hopf bifurcation in an autonomous non-linear feedback system with the time-delay using the frequency}domain approach. Two simple examples of non-linear autonomous delayed systems were presented.…”

Vibration Control of a Cantilever Beam with Time Delay State Feedback

“…Diekmann et al [4] performed a nonlinear analysis for determining Hopf bifurcations in systems of autonomous differential equations. Stepan and Haller [5] considered quasi-periodic oscillations in robot dynamics, and Moiola et al [6] dealt for a more general forced nonlinear system under delay control the response amplitude in the parametric excitation-response curve, while the velocity feedback stabilizes the trivial solution in the frequency-response curve. Thomsen [7] investigated the vibration suppression by using self-arranging 0932-0784 / 06 / 1200-0629 $ 06.00 c 2006 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com mass effects of adding restoring force.…”