Two-cluster solutions in an ensemble of generic limit-cycle oscillators with periodic self-forcing via the mean-field

We study two-cluster solutions of an ensemble of generic limit-cycle oscillators in the vicinity of a Hopf bifurcation, i.e., Stuart-Landau oscillators, with a nonlinear global coupling. This coupling leads to conserved mean-field oscillations acting back on the individual oscillators as a forcing. A reduction to two effective equations makes a linear stability analysis of the cluster solutions possible. These equations exhibit a $\pi$-rotational symmetry, leading to a complex bifurcation structure and a wide variety of solutions. In fact, the principal bifurcation structure resembles that of a 2:1 resonance tongue, while inside the tongue we observe an 1:1 entrainment.

Figure 1

Cluster dynamics in the Stuart-Landau ensemble. Solid lines describe the trajectories of individual oscillators and dots mark their positions in a snapshot. Dashed lines describe the oscillation of the mean-field $\langle W\rangle$. (a) Modulated amplitude clusters for $c_2=-0.6,\nu =0.1$, and $\eta =0.7$. Here, the subgroups perform additional oscillations around their mean-field $\eta exp(-i\nu t)$ given in Eq. (3). (b) Amplitude clusters for $c_2=-0.6,\nu =-1.5$ and $\eta =0.9$. The main difference between the two groups is the difference in radius of their respective limit cycles. The phase shift is much smaller than $\pi$.

Figure 2

Emergence of modulated amplitude clusters in the full ensemble for $c_2=-0.6,\nu =1.2$, and $\eta =0.7$. (a) Modulated amplitude clusters in the original frame. (b) Two limit cycles in antiphase and the fixed point in the origin (red square) in the rotating frame of $w_k$ defined via $W_k=W_0(1+w_k)$. The limit cycles have different radii as the whole population is divided into subgroups with $N_1\ne N_2$.

Figure 3

Cumulative power spectrum for the full system at parameter values $c_2=-0.6,\nu =1.2,\eta =0.7$. The major peaks in this spectrum can be traced back to linear combinations of the Hopf frequency ${\omega }_H$ in Eq. (12) and the frequency of the mean-field oscillation $\nu$ as indicated by the vertical lines [see text and Eq. (15)].

Figure 4

Comparison of the calculated peak frequencies with the frequencies in the full system for $c_2=-0.6$ and $\eta =0.7$. The Hopf bifurcation occurs at ${\eta }_H=1/\sqrt{2}$. (a) Both frequencies $\left|\nu +{\omega }_H\right|$ in blue (dashed) and $\left|\nu -{\omega }_H\right|$ in red (solid) versus $\nu$. Lines describe the results of the linear stability analysis, Eqs. (15), and symbols mark the simulation results. (b) For more details only $\left|\nu -{\omega }_H\right|$ vs. $\nu$.

Figure 5

Coarse bifurcation diagram for the two-groups reduction. Shown is the amplitude $\eta$ of the mean-field oscillation versus its frequency $\nu$ for fixed $c_2=-0.6$. The stable dynamical states are indicated in the figure. The Hopf bifurcation (green) is given by $\eta ={\eta }_H$, the pitchfork (blue) is described by ${\eta }_P^{\pm }$ and the sniper (red) occurs at ${\eta }_{SN}$, see Eqs. (21). The dynamical states along the path A to E are depicted in Fig. 6. The codimension-two points are two Takens-Bogdanov points of $\pi$-rotational symmetry $\left({\mathrm{TB}}_{\pi }^{\pm }\right)$ and two degenerate pitchfork bifurcations (DPF). The details of the bifurcation structure, which have been omitted here, including the unfoldings of the ${\mathrm{TB}}_{\pi }^{\pm }$ points, will be discussed in Sec. 5.

Figure 6

Simulation results for the two-groups reduction in the original frame illustrating the dynamical states along the path A to E in the bifurcation diagram in Fig. 5.

Figure 7

Sketch of the local bifurcation structure around the ${\mathrm{TB}}_{\pi }^{+}$ point with corresponding phase portraits. The involved codimension-one bifurcations are: pitchfork (pf), saddle-node (sn), Hopf (h), saddle-node of infinite period (sniper), and heteroclinic (het). The codimension-two bifurcations are: Takens-Bogdanov ${\mathrm{TB}}_{\pi }^{+}$, degenerate pitchfork (DPF) and saddle-node loop (SNL). Stable fixed points are marked by filled circles and unstable ones by empty circles. Stable limit cycles are drawn with a solid line and unstable limit cycles with a dashed line.

Figure 8

Sketch of the local bifurcation structure around the ${\mathrm{TB}}_{\pi }^{-}$ point with corresponding phase portraits. The involved codimension-one bifurcations are: pitchfork (pf), saddle-node (sn), Hopf (h), saddle-node of infinite period (sniper), saddle-loop (sl), and saddle-node of periodic orbits (snp). The codimension-two bifurcations are: Takens-Bogdanov with symmetry $\left({\mathrm{TB}}_{\pi }^{-}\right)$ and without symmetry (TB), saddle-node loop (SNL), degenerate pitchfork (DPF), and neutral saddle-loop (NSL). Here, a TB and a SNL belonging to different solutions coincide, for details see text. Stable fixed points are marked by filled circles and unstable ones by empty circles. Stable limit cycles are drawn with a solid line and unstable limit cycles with a dashed line. Note that the bifurcation structure in the shaded box is not a result of the continuation as this diverges. It is consistent with the rest of the diagram, but there might be other bifurcations involved, see, e.g., Ref. [18].