Stabilization of unstable rigid rotation of spiral waves in excitable media

Depending on the parameters of two-dimensional excitable or oscillatory media rigidly rotating or meandering spiral waves are observed. The transition from rigid rotation to meandering motion occurs via a supercritical Hopf bifurcation. To stabilize rigid rotation in a parameter range beyond the Hopf bifurcation, we propose and successfully apply a proportional control algorithm as well as time delay autosynchronization. Both control methods are noninvasive. This allows for determination of the parameters of unstable rigid rotation of spiral waves either for a model or an experimental system. Using the Oregonator model for the light-sensitive Belousov-Zhabotinsky reaction as a representative example we show that quite naturally some latency time appears in the control loop, and propose an efficient method to overcome its destabilizing influence.

Figure 1

(a) Isoconcentration lines of the $u$ (solid line) and $v$ field (dash-dotted line) of a rigidly rotating spiral wave. The tip follows the dashed circular trajectory. (b) Tip coordinates are obtained from the intersection between two isoconcentration lines of the $u$ field corresponding to time $t$ (solid line) and time $t+50\mathrm{\Delta }t$ (dotted line).

Figure 2

Tip path patterns obtained under variation of the bifurcation parameter $\varphi$. In the meandering regime ${\varphi }_{cr1}<\varphi <{\varphi }_{cr2}$, rigidly rotating spiral waves are unstable.

Figure 3

(a) The sum of the radii of a meandering tip trajectory $r_1+r_2$ vs the control parameter $\varphi$. The dotted line denotes the radius $r_1$ of unstable rigid rotation in the meandering regime. (b) Dependence of the frequencies ${\omega }_1$ (thick full line) and ${\omega }_2$ (dashed line) on $\varphi$. The dots represent the frequency of unstable rigid rotation in the meandering regime between the two Hopf bifurcations.

Figure 4

Stabilization of rigid rotation under proportional feedback control with tip coordinates defined from the $u$ field ($u_c=0.35$, ${\varphi }_0=0.06$). (a) Trajectory of the spiral tip without feedback ($K=0$, dashed line) and under feedback control ($K=0.005$, full line) which is switched on at $t=25\mathrm{O}.\mathrm{t}.\mathrm{u}$. The cross marks the reference point. (b) Control parameter $\varphi \left(t\right)$ as a function of time; in the stabilized regime the control force $F\left(t\right)$ vanishes.

Figure 5

(a) Tip trajectory under proportional feedback control with tip coordinates defined from the $v$ field. The tip moves to the neighborhood of the reference point (black cross), however, rigid rotation cannot be stabilized. (b) Undamped oscillations of the control force are observed in the asymptotic state.

Figure 6

(a) Successful stabilization of rigid rotation by PFC defining the tip coordinates from the reconstructed activator field $u_{re}$ [Eq. (6) with $\mu =0.05$]. The cross denotes the reference point. (b) Time evolution of the control parameter $\varphi \left(t\right)$ demonstrating the noninvasive character of the control.

Figure 7

$K$-$\delta$ diagram for proportional feedback control for ${\varphi }_0=0.06$. Full circles (small dots) denote successful stabilization (failure of stabilization), the boundary of the control domain determined with steps of 0.0005 in the parameter $K$ is marked by the solid line. Tip definition is based on the $u$ field.

Figure 8

Stabilization of rigid rotation of a spiral wave applying TDAS with tip location defined from the $u$ field. Parameters: $\tau =4.015\mathrm{O}.\mathrm{t}.\mathrm{u}.$, $K=0.001$, and ${\varphi }_0=0.06$. (a) Trajectory of the spiral tip with and without feedback (dashed and full line, respectively). The cross denotes the reference point. (b) Time evolution of the control parameter $\varphi \left(t\right)$.

Figure 9

Control domain for TDAS with tip location defined by the $u$ field for ${\varphi }_0=0.06$ and $\tau =4.015\mathrm{O}.\mathrm{t}.\mathrm{u}.$ Full circles denote successful stabilization, small dots denote failure of stabilization.

Figure 10

(Color online) Control domains for proportional feedback (brighter gray, yellow online) and the TDAS (darker gray, red online) for ${\varphi }_0=0.06$ and $\tau =4.015\mathrm{O}.\mathrm{t}.\mathrm{u}$. Half-logarithmic plot.