TY - JOUR
T1 - Stability analysis of alternating wave solution in a Stuart-Landau system with time delay
AU - Zhang, Shu
AU - Xu, Jian
AU - Chung, Kwok-Wai
PY - 2021/8
Y1 - 2021/8
N2 - In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution.
AB - In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution.
KW - Equivariant Hopf bifurcation
KW - Floquet theory
KW - Stuart-Landau system
KW - Symmetry
KW - Time delay
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85103127712&origin=recordpage
U2 - 10.1016/j.cnsns.2021.105808
DO - 10.1016/j.cnsns.2021.105808
M3 - 21_Publication in refereed journal
VL - 99
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
SN - 1007-5704
M1 - 105808
ER -