Phase-locked scroll waves defy turbulence induced by negative filament tension

Scroll waves in a three-dimensional media may develop into turbulence due to negative tension of the filament. Such negative tension-induced instability of scroll waves has been observed in the Belousov-Zhabotinsky reaction systems. Here we propose a method to restabilize scroll wave turbulence caused by negative tension in three-dimensional chemical excitable media using a circularly polarized (rotating) external field. The stabilization mechanism is analyzed in terms of phase-locking caused by the external field, which makes the effective filament tension positive. The phase-locked scroll waves that have positive tension and higher frequency defy the turbulence and finally restore order. A linear theory for the change of filament tension caused by a generic rotating external field is presented and its predictions closely agree with numerical simulations.

Figure 1

Numerical simulation in Barkley's model showing ordering of scroll wave turbulence by switching on a counterclockwise CPEF at $t=0$ with $E=0.4$ and rotation frequency ${\omega }_e=1.2455$ equal to the natural spiral wave frequency ${\omega }_0$. Filaments shown in yellow. Color-coding shows the recovery variable $v$.

Figure 2

Parameter region of scroll wave turbulence suppression (full circles) as a function of external field amplitude $E$ and normalized frequency ${\omega }_e/{\omega }_0$. Crosses denote failure of ordering turbulence. Initial conditions and model parameters are taken as in Fig. 1.

Figure 3

Parameter region (Arnold tongue) for phase-locking of a 2D spiral wave to the CPEF (full circles) as a function of external field amplitude $E$ and ${\omega }_e/{\omega }_0$. Crosses indicate failure of synchronization.

Figure 4

Change of effective filament tension ${\mathrm{\Gamma }}_1$ of 2D spiral waves under a CPEF. (a) Drift trajectories induced by a small external field, $\left|\right|\vec{e}\left|\right|=0.01$ under different strengths of CPEF. (b) The phase diagram for $E$ vs. ${\omega }_e/{\omega }_0$ of Eq. (1) in 2D for the filament tension in the presence of CPEF. The full circles stand for positive filament tension while the crosses denote the negative one. Data are only shown for phase-locking cases. (c) ${\mathrm{\Gamma }}_1$ vs. ${\omega }_e/{\omega }_0$ for $E=0.20$ (dots) and $E=0.0$ (solid line). The dashed line marks the critical value ${\mathrm{\Gamma }}_1=0$.

Figure 5

The phase diagram of frequency competition for $E$ vs. ${\omega }_e/{\omega }_0$. The full circles represent that the frequency of a 2D CCW spiral wave under a CCW CPEF is higher than that of a 2D CW spiral wave under the same CCW CPEF, while the crosses mean the opposite situation.

Figure 6

Filament tension of phase-locked spiral waves: Comparison of values from electrophoretic drift simulations and linear theory. (a) Resonant case, (b) nonresonant case in the phase-locking regime.