Interconnected Turing patterns in three dimensions

We study numerically the Turing pattern in three dimensions in a FitzHugh-Nagumo-type reaction-diffusion system. We have found that interconnected periodic domain structures such as a gyroid, Fddd, and perforated lamellar structures appear in three dimensions, which never exist in lower dimensions. The stability analysis of these structures is also performed by means of a mode expansion.

Figure 1

Five stationary solutions. The meanings of $L$, $G$, $F$, $P$, $H$, and $S$ are given in the text.

Figure 2

(Color online) (a)–(d) Structural evolution of domains for $\beta =0.04$ and (e) the Bragg positions of the asymptotic structure. For the sake of clarity, the domains in (a) represent the isosurface of $u=0.08$ whereas those in (b)–(d) represent the isosurface of $u=0.05$. (e) The radius of the spheres is proportional to the relative peak intensity. In this figure, Fig. 3b, and Fig. 4b, the Bragg point at the origin is omitted.

Figure 3

(Color online) Fddd structure for $\beta =0.04$ represented by the isosurface of $u=0.05$ and (b) the Bragg positions with $I_k>800$ in Table . The details of (b) are the same as those in Fig. 2e. The Bragg points on the same plane are connected with the line. It is noted that the quasiproper hexagon and the two rectangles are parallel to each other.

Figure 4

(Color online) Perforated lamellar structure for $\beta =0.04$ represented by the isosurface of $u=0.05$ and (b) the Bragg positions with $I_k>750$. The details of (b) are the same as those in Fig. 2.

Figure 5

Phase diagram in the $\gamma \text{−}\beta$ plane. The most stable regions for lamellar, hexagonal. gyroid and BCC are indicated by $L$, $H$, $G$ and $S$, respectively.