From Ballistic to Brownian Vortex Motion in Complex Oscillatory Media

We show that the breaking of the rotation symmetry of spiral waves in two-dimensional complex (period-doubled or chaotic) oscillatory media by synchronization defect lines (SDLs) is accompanied by an intrinsic drift of the pattern. Single vortex motion changes from ballistic flights at a well-defined angle from the SDLs to Brownian-like diffusion when the turbulent character of the medium increases. It gives rise, in nonturbulent multispiral regimes, to a novel “vortex liquid.”

Figure 1

Left: Snapshot of the scalar field $\Delta c_z(\mathbf{r},t)=1/\tau {\int }_0^{\tau }\left|c_z\right(\mathbf{r},t+t^{\prime })-c_z(\mathbf{r},t+t^{\prime }+\tau \left)\right|d{t}^{\prime }$ in the P2 regime for $C=3.5$, $L=256$, and $D=0.4$ (kept fixed throughout). The period of the P1 oscillation is $\tau =5.95$. Black corresponds to $\Delta c_z\approx 0$ and indicates a SDL originating from the spiral core. Right: Magnification of the rectangular region in the left panel showing the SDL (dark grey or red lines) at two times. $\alpha$ is defined as the angle between the spiral core’s trajectory (thick black line) and the attached SDL.

Figure 2

Dependence of the angle $\alpha$ (solid diamonds) and the core velocity, $v$ (circles) on the Rössler parameter $C$. The solid lines are guides to the eye. From left to right, the vertical lines mark the first period-doubling transition to the complex oscillatory regime and the transitions to the turbulent regimes (see text for details).

Figure 3

Dynamics in the turbulent regime close to onset ($C=4.6$, left) and far from onset ($C=5.8$, right) for $L=512$. (a) Snapshots of the $c_x$ field. (b) Trajectories of the spiral core. The black arrow points to the position of the spiral core at the time when the snapshot of $\Delta c_z$ was taken for $C=4.6$ [shown in (c)]. (c) Snapshot of $\Delta c_z$ visualizing the line defects. Note that in the configurations shown three P1 SDLs are attached to the core. The white arrow points to the position of the spiral core.

Figure 4

(a) Mean squared displacement $\langle \left|\Delta \mathbf{r}\right(t){|}^2\rangle$ for different values of $C$ in the turbulent regime. (b) The core diffusion coefficient $D_v$ as a function of $C$.

Figure 5

(a) Snapshot of $\Delta c_z$ showing a typical multispiral configuration in the P1 regime ($C=2.5$, $L=1024$). (b) Same as (a) but in the P2 regime, where P1 SDLs connect spiral cores ($C=3.5$, $L=512$) (c) Same as (b) but for a single pair of spirals connected by a P1 SDL ($L=256$). The direction of the cores’ motion is highlighted by the white arrows. (d) Trajectories of spiral cores leading to the configuration shown in (b). Every 3000 time units the color changes, starting with dark grey/red or black depending on the orientation of the spiral.