Front Reversals, Wave Traps, and Twisted Spirals in Periodically Forced Oscillatory Media

A new kind of nonlinear nonequilibrium patterns—twisted spiral waves—is predicted for periodically forced oscillatory reaction-diffusion media. We show, furthermore, that, in such media, spatial regions with modified local properties may act as traps where propagating waves can be stored and released in a controlled way. Underlying both phenomena is the effect of the wavelength-dependent propagation reversal of traveling phase fronts, always possible when homogeneous oscillations are modulationally stable without forcing. The analysis is performed using as a model the complex Ginzburg-Landau equation, applicable for reaction-diffusion systems in the vicinity of a supercritical Hopf bifurcation.

Figure 1

Front propagation reversal ($n=1,\alpha =0.5$). (a) Standing kinks are found along the dashed line ($\beta =1.8$) in the Arnold tongue. (b) Dependences of velocity $V$ on spatial train period $\lambda$ for $\beta =1.8$ (solid line) and $\beta =5.0$ (dashed line); here $\nu =0.5525$, $B=0.053$. Space-time diagrams showing the behavior of wave patterns after termination of a pacemaker in media with (c) $\beta =5.0$ and (d) $\beta =1.8$, the same other parameters as in part (b); local values of $\mathrm{Re}\eta (x,t)$ are shown in gray scale.

Figure 2

Trapping of kinks and Bloch fronts in the 1D medium. The coefficient $\beta$ is set to $\beta =5$ and decreased to $\beta =1.8$ inside the central region of width 300. Left: Kink trap under the $1:1$ resonance, the same parameters as in Fig. 1b. Right: Bloch front trap under the $2:1$ resonance [$B=0.061$, the same other parameters as in Fig. 1b].

Figure 3

Trapping of kinks and Bloch fronts in the two-dimensional medium. The coefficient $\beta$ is set to $\beta =5$ and decreased to $\beta =1.8$ in the rectangular central region. The medium parameters are the same as in Fig. 1. The system size is $1000\times 1000$. Upper panel: Kink trap in $1:1$ resonance. The snapshots of the spatial distribution of $\mathrm{Re}\eta$ are taken at $t=6000$, $t=19000$ and $t=32000$. Lower panel: Bloch wave trap in $2:1$ resonance. Snapshots the spatial distribution of $\mathrm{Re}\eta$ at $t=2000$, $t=16000$, and $t=46800$ ($B=0.061$).

Figure 4

Rotation of a twisted spiral. Left: Spatial distribution of $\mathrm{Re}\eta$. Right: Position of the spiral at times $t=0$ (solid), $t=340$ (dashed), $t=680$ (dotted). Parameters are $\alpha =4.19$, $\beta =0.992$, $\nu =3.9895$, $B=0.0455$; the system size is $500\times 500$. Numerical integration using the explicit Euler scheme with $\Delta x=0.2$ and $\Delta t=0.0025$.