Stability analysis of alternating wave solution in a Stuart-Landau system with time delay

• Published in: Communications in nonlinear science & numerical simulation Vol. 99; p. 105808
• Main Authors: Zhang, Shu, Xu, Jian, Chung, Kwok-Wai
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2021; Elsevier Science Ltd
• Subjects: Canonical forms; Coordinate transformations; Equivariant Hopf bifurcation; Floquet theory; Hopf bifurcation; Multiscale analysis; Nonlinear systems; Oscillators; Parameters; Stability analysis; Stuart-Landau system; Symmetry; Systems stability; Time delay; Time lag; Waveform analysis; Floquet theory; Equivariant Hopf bifurcation; Stuart-Landau system; Time delay; Symmetry
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.105808

•Large time delay enhances the stability of Alternating Wave Solution (AW).•We combine the method of multiple scales and Floquet theory to study the stability of AW.•The normal form with third order is insufficient to determine the stability of AW.•The mechanism on how nonlinear gains affect the stability of AW is determined. 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.