Fig. 1. Realization sketch of a polarized electric field^[11]: , are two ac electric fields perpendicular to each other, where , are the amplitude and the frequency of the electric field, respectively, and , are the initial phase and the phase difference, respectively.

Fig. 2. Polarized electric fields at different phase differences^[9].

Fig. 3. Clockwise (cw) rotating spiral waves without electric field ^[15]: (a) Highly excitable medium; (b) weakly excitable medium.

Fig. 4. Drifting behaviors of cw spirals under the influence of a polarized electric field^[15]: (a),(b) Drifting behaviors of spirals under the influence of a cw ( ) and a counterclockwise (ccw) ( ) circularly polarized electric fields (CPEFs) with , , , and being the frequency of the spiral waves; (c),(d) dependence of theoretical (lines) and numerical (circles) drift speeds on the phase difference ; (e),(f) dependence of theoretical (lines) and numerical (circles) drift angles on the phase difference . When the drift speed is 0 ( ), the drift angle cannot be defined. (a),(c),(e) Highly excitable medium; (b),(d),(f) Weakly excitable medium.

Fig. 5. Trajectories of spiral tips without control (a)−(e) and under control (f)−(j) of CPEF^[20]. The size in the same column is identical.

Fig. 6. Arnold tongue of the ( )-plane of CPEF^[21]: Lines and circles denote the theoretical and the numerical results, respectively.

Fig. 7. Coherent state out of defect-mediated turbulence accompanied by chiral symmetry breaking^[23]: (a) An initial defect-mediated turbulence state consists of ccw spiral defects (black dots) and cw ones (white dots); (b) coherent state with only ccw spiral waves exists in the asymptotic state when the system is coupled to a ccw CPEF with and ; (c) similar to (b) but with a cw CPEF, and in such a case, only cw spiral waves survive in the system.

Fig. 8. Symmetry breaking of a meandering spiral pair under a ccw CPEF^[23]: (a) ; (b) , , where is the principal frequency of the meandering spiral without the CPEF; (c) E₀ = 0.24, ; (d) dependence of (the frequency of the ccw spiral wave) (full circles) and (the frequency of the cw spiral wave) (open circles) on with ;(e) dependence of (full circles) and (open circles) on with .

Fig. 9. Stabilization of two-armed spiral by CPEF^[27]: (a) With-out external fields; (b) in the presence of a CPEF with , .

Fig. 10. The phase diagram for the effects of CPEF on two-armed spiral^[27]: BU, TS denote the breakup and the stabi-lization regions, respectively, and SS means the region where the electric field is not strong enough to stabilize the two-armed spiral and it decays into two single-armed spirals. The frequency of the single-armed spiral .

Fig. 11. The evolution of a broken plane wave in the subexcitable system without (a)−(c) and with (d)−(f) CPEFs^[11]. , .

Fig. 12. The mechanism analyses for spiral waves sustained by CPEF in subexcitable media^[11], : (a) The sketch of a spiral wave tip submitted to a CPEF; (b) results of varying with ; (c) the comparison of the semi-analytical with the numerical .

Fig. 13. Ordering of scroll wave turbulence by switching on a ccw CPEF at with and rotation frequency equal to the natural spiral wave frequency ^[35]. Filaments are shown in yellow.

Fig. 14. Parameter region of scroll wave turbulence suppression (full circles) as a function of external field amplitude and normalized frequency ^[35]. Crosses denote failure of ordering turbulence.

Fig. 15. Filament tension of phase-locked scroll waves^[35].

Fig. 16. Distribution of the membrane potential induced by CPEF and uniform electric field (UEF)^[40]: (a) CPEF in Luo-Rudy model, , ; (b) UEF in Luo-Rudy model, ; (c) CPEF in Barkley model, , ; (d) UEF in Barkley model, .In Luo-Rudy model, the obstacle size , and in Barkley model, . The red dotted arrows represent the directions of electric fields. The red curved arrows mean CPEFs rotate counterclockwise. The red and the blue regions around obstacles demonstrate de-polarizations and hyper-polarizations, respectively.

Fig. 17. Unpinning the cw rotating anchored spiral by CPEF^[40]: (a) Luo-Rudy model, the frequency of spiral ; , ; CPEF is applied from to . is the initial phase of CPEF relative to x axis; is the initial phase of the anchored spiral front relative to x axis and sets as zero; (b) Barkley model, , , ; CPEF is applied from to . N and N' represent different new waves nucleated by CPEF in different time. S and S' represent the initial anchord spiral and the new free spiral, respectively. White arrows are the propagation directions of waves

Fig. 18. Unpinning scope of CPEF (gray) and UEF (shaded) in Barkley model^[40]: SW, NW, RW and BI regions represent spiral waves, no wave, retracting waves and bi-stability, respectively; for CPEF, , and for UEF, .

Fig. 19. The frequency relations between the circular wave train and CPEF in a two-dimensional quiescent medium^[45]: , 0.22 rad/ms; is the frequency of the circular wave trains, and is the dominant frequency of the spiral turbulence.

Fig. 20. Suppression of spiral turbulence by CPEF^[45]: , : (a) t = 0; (b) t = 1000 ms; (c) t = 1800 ms; (d) t = 2800 ms.

Fig. 21. The frequency relations between the circular wave train and UEF in a two-dimensional quiescent medium^[45]: , ; the pulse duration is ; is the frequency of the circular wave trains, and is the dominant frequency of the spiral turbulence.