Resonant drift of two-armed spirals by a periodic advective field and periodic modulation of excitability

The drift behavior of two-armed spirals induced by periodic advective field and periodic modulation of excitability is investigated. It is shown that the two-armed spirals controlled by periodic advective field and periodic modulation of excitability drift in completely different ways. For periodic advective field, the two tips of the two-armed spiral drift in the same direction and the two-armed spiral is stable. While for periodic modulation of excitability, the two tips drift in the opposite direction and the two-armed spiral splits into two single-armed spirals. Analytical results based on a kinematic theory of rotating spirals in weakly excitable media are consistent with the numerical results.

Figure 1

(Color online) Tip trajectories of two-armed spiral waves. (a) $b=0.245$, (b) $b=0.24$, (c) $b=0.22$, (d) $b=0.2$, (e) the initial phases of two spiral tips ${\psi }_1^0$ and ${\psi }_2^0$ of a counterclockwise rotating two-armed spiral. The tip of the spiral wave is defined as the intersection of contour lines $u=0.3$ and $v=0.3$.

Figure 2

(Color online) (a) The trajectories of the two tips of a clockwise rotating two-armed spiral $\left(\sigma =-1\right)$ controlled by a periodic advective field with ${\omega }_e=2{\omega }_0$, $E_f=0.01$. ${\Theta }_1$ and ${\Theta }_2$ are the drift directions of the two tips, respectively. [(b)–(d)] Variation of the drift direction ${\Theta }_1$ with ${\psi }_1^0$. $\sigma =\pm 1$ represent the chirality of the spiral, $\sigma =1$ for counterclockwise rotating spiral, while $\sigma =-1$ for the clockwise one.

Figure 3

(Color online) The trajectories of the two tips of a clockwise rotating two-armed spiral $\left(\sigma =-1\right)$ controlled by a periodic advective field $\left(E_f=0.01\right)$ with (a) ${\omega }_e={\omega }_0$, (b) ${\omega }_e=1.5{\omega }_0$, (c) ${\omega }_e=2.5{\omega }_0$, (d) ${\omega }_e=3{\omega }_0$. The total time of the simulations is 500.

Figure 4

(Color online) (a) The trajectories of the two tips of a clockwise rotating two-armed spiral when the excitability of the medium is modulated periodically with ${\omega }_b={\omega }_0$, $b_f=0.0035$. ${\Theta }_1$ and ${\Theta }_2$ are the drift directions of the two tips, respectively. The two tips drift in the opposite direction and the two-armed spiral splits into two single-armed spirals. (b) The trajectories of the two tips of a two-armed spiral with resonant modulation with $b_f=0.0027$. Two-armed spiral cannot be split into two single-armed spirals. [(c) and (d)] Variation of ${\Theta }_1$ with ${\psi }_1^0$. The parameter are $b_f=0.003$, ${\omega }_b={\omega }_0$.

Figure 5

(Color online) The trajectories of the two tips of a clockwise rotating two-armed spiral $\left(\sigma =-1\right)$ when the excitability of the medium is modulated periodically $\left(b_f=0.003\right)$ with (a) ${\omega }_b=0.5{\omega }_0$, (b) ${\omega }_b=1.5{\omega }_0$, (c) ${\omega }_b=2{\omega }_0$, (d) ${\omega }_b=2.5{\omega }_0$. The total time of the simulations is 500.

Figure 6

(Color online) [(a) and (b)] Dependence of the drift of a single-armed spiral by a periodic advective field (${\omega }_e=2{\omega }_0$, $E_f=0.01$) on the initial phase of the spiral. [(c) and (d)] Dependence of the drift of single-armed spirals by periodic modulation of excitability (${\omega }_b={\omega }_0$, $b_f=0.003$) on the initial phase of the spiral.