Complex-anisotropy-induced pattern formation in bistable media

A construct of anisotropy in bistable media is adopted to characterize the effects of anisotropy on pattern formation by means of anisotropic line tension. A velocity curvature relation is further derived to account for the anisotropic wave propagations. Stability analysis of transverse perturbations indicates that a sufficiently strong complex anisotropy can induce dynamical instabilities and even lead to a breakup of the wave patterns. Numerical simulations show that complex anisotropy can induce rich spatiotemporal behaviors in bistable media. The results of analysis and simulations demonstrate that this method successfully incorporates complex anisotropy into the reaction diffusion model and has general significance.

Figure 1

(Color online) Under the choice $\gamma =1+0.04\cos \left(4\theta \right)$, the velocity of planar $(\kappa =0)$ fronts at different angles. The Ising fronts exist in all directions, while the Bloch fronts exist in narrow regimes around $\theta =0$, $\theta =\pi ∕2$, and $\theta =\pi$.

Figure 2

(Color online) NIB bifurcation and planar front transverse instability boundaries in the $(\epsilon ,\delta )$ plane for the anisotropic $(\tilde{\gamma }=1+0.03\cos 4\theta )$ and nonsymmetric $(a_0=-0.1)$ systems. The thick curve denotes the front bifurcation. The thin lines present the transverse instability boundaries of Ising fronts with ${\delta }_I$ and Bloch fronts with ${\delta }_B$. The dashed lines present wave propagation along the $\theta =3\pi ∕4$ direction; the dashed lines present wave propagation along the $\theta =0$ direction. The parameter $a_1=2.0$.

Figure 3

(Color online) The “square-shaped” contour of stationary patterns $(C_n=0)$ with complex anisotropy $\tilde{\gamma }=\sqrt{q_1\left(\theta \right){\cos }^2\theta +q_2\left(\theta \right){\sin }^2\theta }$ with $q_1\left(\theta \right)=2$ and $q_2\left(\theta \right)=1+b{\sin }^{2}2\theta$. (a) The smooth contour for $b=0.5$ and (b) the contour with the outer loops (“swallow tails”) for $b=1.5$. Such loops should be cut off because of no physical meaning.

Figure 4

(Color online) Polygon-shaped target waves induced by complex anisotropy with parameters corresponding to the stable Bloch fronts. (a) The elliptic target wave under the choice of $\gamma =\sqrt{q_1\left({\varphi }_1\right){\cos }^2{\varphi }_1+q_2\left({\varphi }_1\right){\sin }^2{\varphi }_1}$, with $q_1\left({\varphi }_1\right)=2$, $q_2\left({\varphi }_1\right)=1+0.5{\sin }^2{\varphi }_1$, and ${\varphi }_1=\varphi +\pi ∕4$. (b) The triangular target wave with $\gamma =1+0.1\cos \left(3\varphi \right)$. (c) The rectangular target wave with the same anisotropy form to (a) and $q_1\left({\varphi }_1\right)=2$, $q_2\left({\varphi }_1\right)=1+0.5{\sin }^{2}2{\varphi }_1$, and ${\varphi }_1=\varphi$. Parameters: $\delta =1.0$, $\epsilon =0.02$

Figure 5

(Color online) The breakup of a target wave is due to collision with the boundary of the system for complex anisotropy $\gamma =1+0.07\cos \left(4\varphi \right)$. The initial condition is the same as that in Fig. 3. (a) $t=1$, (b) $t=8$, and (c) $t=20$. Parameters: $\delta =1.0$, $\epsilon =0.02$.

Figure 6

(Color online) The velocity-curvature relation for different orientations of wave propagation. Parameters: $a_0=-0.1$, $a_1=2.0$, $\alpha =0.09$, $m=4$, $\delta =1.0$, $\epsilon =0.02$.

Figure 7

(Color online) The breakup of a target wave is induced by the strong anisotropy $\gamma =1+0.09\cos \left(4\varphi \right)$ and develops into spatiotemporal chaos finally. (a) $t=1$, (b) $t=3$, and (c) $t=170$. Parameters: $\delta =1.0$, $\epsilon =0.02$.

Figure 8

(Color online) The development of a pentagon-shaped spiral wave with $\gamma =1+0.035\cos \left(5\varphi \right)$ and parameters corresponding to the stable Bloch fronts. (a) $t=1$, (b) $t=6$, and (c) $t=50$. Parameters: $\delta =1.0$, $\epsilon =0.025$.

Figure 9

(Color online) Pattern formation from a single stripe with transverse perturbation in its middle part [see Fig. 10a] in the unstable Ising regimes. (a) Labyrinthine pattern in the isotropic case, (b) comb-shaped stripes with $\gamma =1+0.04\cos \left(4\varphi \right)$, and (c) directional stripes with $\gamma =\sqrt{2{\cos }^2\varphi +q_2\left(\varphi \right){\sin }^2\varphi }$ and $q_2\left(\varphi \right)=1+0.5{\sin }^2\varphi$. Parameters: $\delta =4.0$, $\epsilon =0.045$.

Figure 10

(Color online) Under the choice of $\gamma =1+0.1\cos \left(4\varphi \right)$, the development of directional stripes from a single stripe with transverse perturbation in its middle part. (a) $t=1$, (b) $t=50$, and (c) $t=420$. Parameters: $\delta =4.0$, $\epsilon =0.045$.