In these experiments and simulations, we show that the Kibble-Zurek mechanism (which is a causal arisen mechanism; for more information, see the topic Cosmology in the laboratory) holds in a weak secondary subcritical symmetry-breaking bifurcation of a quasi-1D system of non-locally coupled oscillators. Subcriticality is responsible for the coexistence of different coherent patterns under a fixed value of the control parameter showing bistability, or even multistability.

The complex 1D quintic Ginzburg-Landau equation associated to each oscillator allows us to describe the amplitude-phase portrait of a secondary subcritical bifurcation for a real network of 100 non-locally coupled convective oscillators.

The convective oscillators are thermo-convective cells that emerge in a thin film of fluid owing to a localized quasi-1D heating. For a given depth of the fluid layer, the control parameter is the vertical temperature difference. The stability diagram of this system exhibits a rich phenomenology ('chimera states', pulses, fluctuating and stationary fronts, traveling waves, alternating patterns, topological defects) [M.A. Miranda et al., Phys. Rev. E 78 (2008) 046305][M.A. Miranda et al., Phys. Rev. E 78 (2009) 046201][M.A. Miranda et al., Int. J. Bifurcation and Chaos 20 (2010) 835][M.A. Miranda, PhD thesis (2009) Univ. of Navarra].

The Kibble-Zurek mechanism is responsible for the frozen dynamics, at the core of a secondary symmetry breaking bifurcation, which consists in a defect trapping phenomenon at the synchronization 1D-front that connects the initial stationary pattern (multicellular pattern) and the oscillatory one (anti-phase/in-phase pattern) [M.A. Miranda et al., Int. J. Bifurcation and Chaos 22 (2012) 1250165][M.A. Miranda et al.Phys. Rev. E 87 (2013), 032902]. Below, a typical space-time diagram obtained from the experiment and from the numerical simulation.

This phenomenon is driven by quenching the system towards the less symmetric state where the perturbations have not enough time to become damped due to the constraints imposed by the 'horizon' (limiting speed of sound in our system), and therefore, become frozen in the form of trapped defects. The quenched dynamics is the response to different crossing rates when the system, which is initially settled close and below the threshold, is driven throughout the aforementioned bifurcation. Whereas, the correlation length of the system is given by the self-correlation of the synchronization 1D-fronts. Below, we show the fit of the correlation length of the 1D-front vs the crossing rate to a power law of the experimental and the numerical results.

In the framework of complex systems, we are dealing with the quenched synchronization process of a geometrical network of non-locally coupled oscillators throughout a secondary bifurcation. We importantly show in this report that:

• The reduction of degrees of freedom from the continuum to the convective oscillators make the effective interaction non-local.

• A network of supercritical oscillators non-locally coupled [M.A. Miranda et al., Phys. Rev. E 78 (2009) 046201][M.A. Miranda et al., Int. J. Bifurcation and Chaos 20 (2010) 835] leads to a globally subcritical bifurcation (multi-stability).

• We are able to model the non-local interaction by a coupling à la Kuramoto, while keeping the non-spatial part as a standard complex quintic Ginzburg-Landau equation. Thus, phase synchronization and amplitude instability is obtained.

• The Kibble-Zurek mechanism prevails against the effect of non-locality for a critical value of the crossing rate, μ_c. At faster values of the crossing rate μ≥μ_c the Kibble-Zurek mechanism is tested in accordance with the experimental results [M.A. Miranda et al., Int. J. Bifurcation and Chaos 22 (2012) 1250165][M.A. Miranda et al. Phys. Rev. E 87 (2013), 032902]. In summary, as the interaction range (non-locality) is increased, the crossing rate has to be increased in order to de-correlate (de-synchronize), according to the Kibble-Zurek mechanism, distant regions of the network.

The study of Kibble-Zurek mechanism in our system paves the way for a better understanding of frozen dynamics in synchronization processes in nature (differentiation mechanisms in morphogenesis beyond the synchronization front) and in experiments (where non-locality triggers global synchronization towards 'chimera states', or multi-stable states), where in agreement with the Kibble-Zurek mechanism, the correlation length of the system scales with the quench rate.

• • M.A. Miranda et al. Phys. Rev. E 87 (2013), 032902.

This work has been partly supported by the Spanish contract No. FIS2008-01126, FIS2011-24642, and by the Gobierno of Navarra (Depto. de Educación).