Frozen dynamics and synchronization through a secondary symmetry-breaking bifurcation

We show evidence of the frozen dynamics (Kibble-Zurek mechanism) at the transition one-dimensional (1D) front of an extended 1D array of convective oscillators that undergo a secondary subcritical bifurcation. Results correspond to a global synchronization process from nonlocal coupling between the oscillating units. The quenched dynamics exhibits defect trapping at the synchronization front according to the Kibble-Zurek mechanism, predicted for condensed matter systems. A stronger subcriticality prevents the fronts from freezing defects during the quenched transitions. A synchronization model of supercritical oscillating units is proposed to explain differentiation mechanisms in morphogenesis above a critical crossing rate when the frequency of the individual oscillators becomes coherent. The phases of such oscillators are spatially coupled through a Kuramoto-Battogtokh term that leads to the experimentally observed subcriticality. As a consequence, we show that the Kibble-Zurek mechanism overcomes non-locality of a geometrical network above a critical crossing rate.

Figure 1

(a) Sketch of the 1D array of coupled oscillators (each oscillator is given by a convective cell) whose coherent phases describe a long wavelength profile. Bottom: A shadowgraphy image of the convective cell corresponding to the oscillatory cells in the antiphase (ALT) pattern coexisting with latent oscillatory cells (ST pattern). The average wavelength of the stationary pattern is approximately 4.6 mm and becomes doubled in the ALT. (b) Sketch of the two subsequent secondary subcritical bifurcations from the ST pattern towards the ALT pattern. Dashed arrows depend on the crossing rate value. (c) Quasistatic threshold ($\epsilon =0$) at a secondary symmetry-breaking instability from a multicellular pattern (${\epsilon }_i$ is the initial control parameter value) towards an oscillatory pattern.

Figure 2

Experimental spatiotemporal diagrams (the average recording time is 15 min along with roughly 50 convective cells) of quenched transitions for (a) a weak subcritical secondary bifurcation (at $d=7.5$ mm, from 36 ${}^{\circ }$C towards 52 ${}^{\circ }$C), where two topological defects are trapped at the transition 1D front, and (b) a strong subcritical secondary bifurcation (at $d=7$ mm, from 38 ${}^{\circ }$C towards 50 ${}^{\circ }$C) with a nonhomogeneous 1D front. The transition 1D fronts are defined at $e^{-1}$ of the maximum critical amplitude.

Figure 3

(a) Experimental results on the correlation length vs the crossing rate for a secondary weak subcritical bifurcation ($d=$ 7.5 mm, circles) and for a strong secondary subcritical bifurcation ($d=$ 7 mm, squares). The dashed line for $\mu \ge {\mu }_c$ (where ${\mu }_c$ is the critical crossing rate for the most homogeneous front) is fitted to the power law $\xi \sim {\mu }^{-2.8\pm 0.1}$ for the secondary weak subcritical bifurcation. (b) Numerical results on the correlation length vs the crossing rate. The dashed line for $\mu \ge {\mu }_c$ (where ${\mu }_c$ is the critical crossing rate for the most homogeneous front) is fitted to the power law $\xi \sim {\mu }^{-2.2\pm 0.1}$.

Figure 4

Spatiotemporal diagrams obtained from the numerical simulation at different crossing rates (arbitrary units): (a) $\mu =0$ (quasiadiabatic transition), (b) $\mu ={\mu }_c=6$, (c) $\mu =10$.

Figure 5

On the left spatiotemporal diagrams from the numerical simulations with the corresponding 1D front, and on the right a mean-field approach of the synchronization process beyond the threshold in panel (a) at $\mu =6$ and in panel (b) at $\mu =8$.

Figure 6

Spatiotemporal diagram obtained from the numerical simulation at ${\mu }_c=6$. The 1D front, which is defined at $e^{-1}$ of the maximum amplitude, splits the diagram into the multicellular pattern and the new antiphase pattern.