Localized spatiotemporal reaction-diffusion patterns on a line and a disk arising from a subcritical finite wavenumber Hopf instability

Abstract

Spatiotemporal localized and extended structures associated with a subcritical finite wavenumber Hopf bifurcation are studied in the Purwins model (a three-variable FitzHugh-Nagumo version). Steady and time-dependent numerical continuation procedures are used to investigate snaking behavior of localized standing and traveling waves on the real line, and the results are corroborated using weakly nonlinear theory. The results shed light on the origin of so-called jumping oscillons and the organization of a nontypical homoclinic snaking structure of traveling pulses. The computations are extended to moderate size disks and used to identify wall-attached spots that travel along the disk boundary as well as wall-attached spots that oscillate in place and wall-attached jumping oscillons. The one-dimensional results are shown to be useful in interpreting the two-dimensional results. Domain-filling and mixed structures are also studied, demonstrating the variety of extended and localized states that emerge in two-space dimensions, ranging from periodic to disordered. The latter are potentially important for observations of waves in far-from-equilibrium media, such as those often observed in cell biology.

Figures (21)

• Figure 1: Space-time profiles of a jumping oscillon (1JO, left panel), a bound pair of a jumping oscillon and a traveling pulse (1JP-1TP, middle panel) and a bound pair of jumping oscillons (2JO, right panel). Reused from knobloch2021origin.
• Figure 1: Bifurcation diagram (left panels) obtained via continuation with NBCs on a domain of length $L{=}4\cdot 2\pi/q_c\simeq 89.76$, showing branches of uniform standing waves (SWs, magenta) and spatially localized standing waves (LSWs, brown) in terms of the norm $\|U\|_2$ and the temporal period $\tau$, together with selected space-time plots at locations (i)-(v). Here, $x=0$ indicates the center of the LSW; the solutions must be reflected in $x{=}0$ to obtain the full LSW solution. The dotted white lines in panels (i) and (v) show that the SW oscillate in phase at every location while this is no longer the case for the LSW where the outer regions lag behind the center. The corresponding results from the amplitude equations are shown in dashed lines: magenta line is from \ref{['eq:SW_ampltds']}, while the orange line is obtained from \ref{['eq:clSW']} (i.e., with $S_g=0$) and the brown line is from continuation of \ref{['eq:final_ampltd']}; in the top left panel the orange and the brown lines lie one on top of one another. For reference, we also plot the TW branch from \ref{['eq:TW_ampltds']} [dashed blue line, see also Fig. \ref{['fig:appTW']}]. Thick (thin) lines indicate stable (unstable) states.
• Figure 1: Bifurcation diagram obtained via continuation on a disk of radius $R=20$ with $D_w=50$, showing branches of wall-attached rotating spot trains [RSTs, blue, state (i) at $k_1=-8.354$], rotating spots [1RSs, green, state (iii) at $k_1=-8.125$] and two short PO segments: JO [orange, state (iv) at $k_1=-7.148$] and a pulsating filament [also orange, state (v) at $k_1=-7.762$, in corotating frame]. Bulk rotating spots set in very close to $k_{1c}$ [2RSs, blue-green, state (ii) at $k_1=-7.147$]. Thick lines (green and orange) indicate linear stability: the 1RS state is stable between the fold on the left and a Hopf bifurcation at $k_1\equiv k_{1H}\simeq -7.798$ and transfers stability to the filament state (v) at $k_1\equiv k_{1H}$. All states rotate in a clockwise direction as indicated in (i). In (iv), we show the space-time evolution along the disk perimeter ($\xi\in [-\pi R,\pi R)$) of a modulated 1RS state in the corotating frame arising from a Hopf bifurcation of the 1RS state (green square).
• Figure 1: Bifurcation diagram of TWs obtained via continuation using AUTO (solid line) and the solution of amplitude equations (dashed line) in terms of $u_{\rm max}$ reconstructed from \ref{['eq:TW_ampltds']} and \ref{['eq:slowamp']}. The inset shows the respective speeds. A similar comparison between numerically computed SWs and LSWs and the amplitude equations is shown in Fig. \ref{['fig:fig3']}.
• Figure 2: Dispersion relations $\sigma(q)$ summarizing the linear stability properties of $\mathbf{U}_*$ for various values of $k_1$, from left to right: $k_1=-7.9<k_{1c}$ (stable), $k_1=-7.585\simeq k_{1c}$ (critical), $k_1=-7.2>k_{1c}$ (unstable). The real (imaginary) parts of $\sigma$ are indicated in blue (red) on the left (right) axes. Note that the decaying ${\rm Re}\,\sigma$ from $q=0$ is in fact accompanied by ${\rm Im}\, \sigma=0$.