Homoclinic snaking in bounded domains

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of “snaking without bistability,” recently observed in simulations of binary fluid convection by Mercader et al. Phys. Rev. E80, 025201 (2009).

Figure 1

(Color online) (a) Bifurcation diagram showing the equivalent of homoclinic snaking in the Swift-Hohenberg equation with $b_2=1.8$ on a periodic domain with period $\Gamma =62$. The snaking branches $L_0$ and $L_{\pi }$ emerge from $P_{10}$ in a secondary bifurcation (○) at small amplitude and terminate on the same branch in a secondary bifurcation (○) near the saddle node. Other spatially periodic branches are also present but for clarity are not shown. [(b) and (c)] Sample profiles from the ${\varphi }^{\prime }=0$ and ${\varphi }^{\prime }=\pi$ branches near the upper end of the snaking branches. [(d) and (e)] Sample profiles from the $\varphi =0$ and $\varphi =\pi$ branches on the lower part of the snaking branches. In this case the $L_0$ branch connects the $\varphi =0$ and ${\varphi }^{\prime }=0$ solutions, and the $L_{\pi }$ branch connects the $\varphi =\pi$ and ${\varphi }^{\prime }=\pi$ solutions. After [12].

Figure 2

Bifurcation diagram showing the equivalent of homoclinic snaking with Neumann boundary conditions on a domain of length $2L=40$. Only even-parity states are shown. Pitchfork bifurcations are denoted by ○. Solution profiles $u\left(x\right)$ at the saddle nodes (a)–(d) are shown in Fig. 4.

Figure 3

Sample large amplitude periodic states $u\left(x\right)$ on (a) $P_6$ and (b) $P_5$ with Neumann boundary conditions on a domain of length $2L=40$, both at $r=0$.

Figure 4

Sample two-pulse profiles $u\left(x\right)$ at the saddle nodes along the figure-eight isola in Fig. 2.

Figure 5

(Color online) Neutral curves for (a) $\beta =0$ and (b) $\beta =0.5$ corresponding to even (red solid) and odd (blue dashed) eigenfunctions.

Figure 6

Bifurcation diagram showing the equivalent of homoclinic snaking with Robin boundary conditions (2) and $\beta =0.5$ on a domain of length $2L=40$. Only the $\varphi =0$ branch $S_{6,0}$ is shown. Solution profiles $u\left(x\right)$, at the labeled points, are shown in Fig. 8.

Figure 7

(Color online) (a) With Robin boundary condition (2) the bifurcation at $S_6$ becomes transcritical (solid dot): the $\varphi =\pi$ branch $S_{6,\pi }$ now bifurcates supercritically but undergoes a saddle-node bifurcation close to threshold before turning toward negative values of $r$, while $S_{6,0}$ bifurcates subcritically. Thus the breaking of the hidden reflection symmetry results in an apparent splitting of the $P_6$ branch. (b) The same but on a larger scale, showing $S_{6,0}$ (dashed green) and $S_{6,\pi }$ (solid red) together with the two branches that become disconnected from them as a result of the broken hidden symmetry. Parameters: $\beta =0.5$ and $2L=40$.

Figure 8

Solution profiles $u\left(x\right)$ at successive saddle nodes along the $\varphi =0$ branch $S_{6,0}$ from small to large amplitude as indicated in Fig. 6. The final panel shows the $\varphi =0$ state at $r=0.12997$. Parameters: $\beta =0.5$ and $2L=40$.

Figure 9

The $S_{6,\pi }$ branch for $\beta =0.5$ on a domain of length $2L=40$. The branch connects to $S_7$ via a complicated sequence of saddle-node bifurcations.

Figure 10

Solution profiles $u\left(x\right)$ at successive saddle nodes along the $S_{6,\pi }$ branch in Fig. 9 from small to large amplitude and then back toward small amplitude along the $S_7$ branch. Parameters: $\beta =0.5$ and $2L=40$.

Figure 11

The isola of two-pulse $\varphi =\pi$ states computed for $\beta =0.5$ and $2L=40$. A similar isola of $\varphi =0$ states is omitted.

Figure 12

Solution profiles $u\left(x\right)$ at successive saddle nodes along the two-pulse isola in Fig. 11 when $\beta =0.5$ and $2L=40$.

Figure 13

A blow-up of the region on the $S_{7,0}$ branch (Fig. 9) showing a pair of saddle-node bifurcations created from an imperfect bifurcation. Solution profiles (p) and (q) are shown in Fig. 10.

Figure 14

Bifurcation diagram for $\beta =0.5$ and $L=20$ showing the large amplitude $\varphi =\pi$ branch. This branch becomes disconnected from the smaller amplitude $\varphi =\pi$ branch $S_{6,\pi }$ shown in Fig. 9 near (b) but connects to a subsequent primary bifurcation at $S_5$.

Figure 15

Detail of Fig. 14.

Figure 16

Solution profiles $u\left(x\right)$ at successive saddle nodes in Fig. 14 (and Fig. 15), starting from the large amplitude $P_6$ state (a) and finishing at the last saddle node $\left(T\right)$ before the primary bifurcation at $S_5$.

Figure 17

(Color online) Bifurcation diagram for the case $\beta =0$, $L=20$. The pattern mode $A_6$ (green dashed) is shown together with the lowest of a stack of figure-eight isolas (red solid). Sample solution profiles are shown in Figs. 18, 19.

Figure 18

Solution profiles $u\left(x\right)$ on the branch $A_6$ in Fig. 17 at (i) small amplitude, (ii) the saddle node and (iii) large amplitude. The mean of the solution increases monotonically with amplitude.

Figure 19

Solution profiles $u\left(x\right)$ at each of the saddle nodes on the figure-eight isola shown in Fig. 17.

Figure 20

Bifurcation diagram for $\beta =0.5$ and $L=20$ showing the branch $A_6$ of nonsymmetric states. The branch is continuous and follows all parts of the snake-and-ladders structure of the pinning region, and does so several times. The filled circle at large amplitude denotes the point at which continuation of the branch was stopped.

Figure 21

Solution profiles $u\left(x\right)$ at the five saddle nodes visible (right to left) in the inset in Fig. 20. The first three are related to distinct two-pulse states present with NBC.

Figure 22

Bifurcation diagram for $\beta =0.5$ and $L=20$ showing the branch $A_5$ of nonsymmetric states until the branch terminates on the $S_{5,0}$ branch at (○). This point is also marked on Fig. 14 for comparison. Solution profiles at each saddle node are shown in Fig. 23.

Figure 23

Solution profiles $u\left(x\right)$ at successive saddle nodes in Fig. 22 starting near the first bifurcation and ending at the termination of $A_5$ on the even-parity branch $S_{5,0}$.

Figure 24

Bifurcation diagrams with boundary conditions (5) for (a) $\beta =0$, $L=35$, (b) $\beta =1$, $L=35$, and (c) $\beta =1$, $L=40$. Solid (dashed) lines indicate $\varphi =0$, $\pi$ states, respectively. Only even-parity states are shown.

Figure 25

Bifurcation diagrams with the boundary conditions (5) for $\beta =10$ and $L=35$. (a) $\varphi =0$ and (b) $\varphi =\pi$ (isola only).

Figure 26

Solution profiles $u\left(x\right)$ for the boundary conditions (5) with $\beta =10$ and $L=35$ at the saddle nodes indicated in Fig. 25a.