Snapshots of the frozen states in excitable media within the parameter regions of rigidly rotating spiral and meandering spiral, respectively. The parameters are (a) and (b) . The positions of shocks are emphasized by small open circles. In (a), the theoretic positions from Eq. (5) (solid line) agree with the real positions of shocks (open circles) well.
Evolution of the pattern of a pair of rigidly rotating spirals at and , respectively, showing the frozen structure of shocks. is the period of one complete tip trajectory. . In (a), the theoretic result from Eq. (5) considering the chiralities effect (solid line) fits the position of shocks (circles) well, while the hyperbola from the BHO theory without considering the chiralities effect (dashed line) shows a big deviation. In this calculation, , , , , and .
The same as Fig. 2 with within the parameter region of meandering spiral considered instead. is the period of one complete cycloid. Both of the analytic positions from Eq. (5) (solid lines) and the BHO theory (dashed lines) deviate from the real position (circles).
(a) Schematic show of the characterization of shock dynamics of frozen state in excitable media. (b) Illustration for the formation of a shock in frozen state in oscillatory media. [(c) and (d)] Fits of Archimedean spirals [gray lines (red online)] to the simulation results (black dots) for a rigidly rotating spiral and a meandering spiral, respectively. The parameters are (c) and (d) . For the rigidly rotating spiral, the approximation of an Archimedean spiral starting from the center of tip trajectory is nice. For the meandering spiral, however, the approximation is never applicable.
[(a) and (b)] The plots of as a function of time for different parameter ’s. [(c) and (d)] The plots of and the periods and vs , respectively. The intermediate vertical dashed line at denotes the parameter for a modulated traveling wave, dividing the two gray regions for the inward-petal meandering spiral and outward-petal meandering spiral.
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