Bifurcation analysis of delay-induced periodic oscillations (Q847194)

The paper provides a detailed numerical bifurcation analysis of the following system with time-delay and cubic nonlinearity \[ \frac{dz}{dt} = (\alpha + i \beta) z -z|z|^2 +z(t-\tau), \] where \(z(t)\) is a complex variable and \(\alpha\) and \(\beta\) are real parameters. As shown in [\textit{M. Wolfrum} and \textit{S. Yanchuk}, Phys. Rev. Lett. 96, 220201 (1997)] this system undergoes oscillatory (Eckhaus) instability with a number of periodic solutions appearing as the parameter \(\alpha\) increases over \(-1\). The present research uses the DDE-Biftool software [see \textit{D. Roose, T. Luzyanina, K. Engelborghs} and \textit{W. Michiels}, Lecture Notes in Computational Science and Engineering 38, 167--181 (2004; Zbl 1065.34077)] and investigates the bifurcation mechanisms behind this.