Linked and knotted chimera filaments in oscillatory systems

While the existence of stable knotted and linked vortex lines has been established in many experimental and theoretical systems, their existence in oscillatory systems and systems with nonlocal coupling has remained elusive. Here, we present strong numerical evidence that stable knots and links such as trefoils and Hopf links do exist in simple, complex, and chaotic oscillatory systems if the coupling between the oscillators is neither too short ranged nor too long ranged. In this case, effective repulsive forces between vortex lines in knotted and linked structures stabilize curvature-driven shrinkage observed for single vortex rings. In contrast to real fluids and excitable media, the vortex lines correspond to scroll wave chimeras [synchronized scroll waves with spatially extended (tubelike) unsynchronized filaments], a prime example of spontaneous synchrony breaking in systems of identical oscillators. In the case of complex oscillatory systems, this leads to a topological superstructure combining knotted filaments and synchronization defect sheets.

Figure 1

Chimera nature of knots ($\alpha =0.8, R=8, L=200$). (a) Snapshot of the Hopf link shows the unsynchronized phase $\theta (\mathbf{r},t)$ on the isosurface of spatially smoothed angular frequency $\tilde{\omega }(\mathbf{r},t)$. (b) Mean angular frequency distribution of $\langle \omega \left(\mathbf{r}\right)\rangle$, averaged over approximately 10 periods. The tail to the right corresponds to unsynchronized oscillators. (c) Plot of 2D cross section of $\theta \left(\mathbf{r}\right)$ showing the chimera and spiral wave properties. A vertical cut through both rings in panel (a), showing four chimera cores. (d) A cut through the far edge of a ring. (e) Slicing of a ring, showing a ring chimera and two chimera cores of the other ring.

Figure 2

Phase diagram of the Kuramoto model in 3D as a function of $\alpha$ with nonlocal coupling $L\gg R\gg 1$. Various knots exist between ${\alpha }_K$ and ${\alpha }_0$. Top panel shows stable Hopf link (left) and trefoil (right) as examples. The plots are similar to those in Fig. 1, but smoothed phases $\tilde{\theta }\left(\mathbf{r}\right)$ are used instead. Shadows on the walls correspond to perpendicular projections of the structures.

Figure 3

Time evolution of Hopf links outside their stability regimes near ${\alpha }_K$ and ${\alpha }_0$, respectively. (a) $\alpha <{\alpha }_K$. After $\alpha$ is decreased slightly from above ${\alpha }_K$ to below, the two rings of the Hopf link merge, resulting in a single ring which eventually vanishes. (b) $\alpha >{\alpha }_0$. After $\alpha$ is tuned slightly from below ${\alpha }_0$ to above, one of the rings grows until it collides and reconnects, resulting in a turbulent-like pattern.

Figure 4

(a) The tolerable phase noise level for Hopf links (HL) and trefoils (T). $f$ quantifies the fraction of structures persisting after $t=5000$ ($\approx 450$ scroll wave rotations) under noise intensity $D$. Each point corresponds to an ensemble of 10 realizations. Here, $\alpha =0.8, R=4$, and $L=100$. Inset: $D^{*}$ denotes the point of $f=0.5$ for a Hopf link as a function of $R$ using $\alpha =0.8, L/R=16$. (b) Phase diagram showing the stable regime of Hopf links for kernel $G_1$ with $L\gg R\gg 1$. The red dot marks the triple point between all three phases where ${\alpha }_K={\alpha }_0$. The error bars account for different $R\ge 4$ and $L$ up to $L=300$. Note that $1/n_1=0$ corresponds to the top-hat kernel $G_0$ used in Fig. 2.

Figure 5

(a) CGLE $\left[\right(a,b)=(1,0),K=0.1]:$ Comparison of minimum separation between rings in a Hopf link, $d_{min}$, and a measure of spontaneous fluctuations in the ring shape, $\xi$ (see the SM [39] for details). Bars indicate the 99% range. (b) Rössler model: Topological superstructure of a chimera Hopf link with an attached sychronization defect sheet in the period doubled regime (see the SM [39] for more viewpoints and dynamics).