Nontrivial twisted states in nonlocally coupled Stuart-Landau oscillators

A twisted state is an important yet simple form of collective dynamics in an oscillatory medium. Here we describe a nontrivial type of twisted state in a system of nonlocally coupled Stuart-Landau oscillators. The non-trivial twisted state (NTS) is a coherent traveling wave characterized by inhomogeneous profiles of amplitudes and phase gradients, which can be assigned a winding number. To further investigate its properties, several methods are employed. We perform a linear stability analysis in the continuum limit and compare the results with Lyapunov exponents obtained in a finite-size system. The determination of covariant Lyapunov vectors allows us to identify collective modes. Furthermore, we show that the NTS is robust to small heterogeneities in the natural frequencies and present a bifurcation analysis revealing that NTSs are born or annihilated in a saddle-node bifurcation and change their stability in Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The latter can lose stability in a supercritical Hopf bifurcation, leading to a modulated 2-NTS.

Automated summary

Twisted state is an important and simple form of collective dynamics in oscillatory medium, characterized by inhomogeneous profiles of amplitudes and phase gradients. In this study, we investigate a non-trivial twisted state in a system of nonlocally coupled Stuart-Landau oscillators using various methods including linear stability analysis, Lyapunov exponents, and covariant Lyapunov vectors. We show that the non-trivial twisted state is robust and can be born or annihilated in saddle-node bifurcations and change stability in Hopf bifurcations.