Figure 1

Plot of wave front speed $c$ as a function of $\epsilon$ for fixed $\beta$ and $\kappa =0.25$. Stable (unstable) branches are shown as solid (dashed) curves. (a) If $2\kappa (1+\beta )=1$ then there exists a stationary front for all $\epsilon$; at a critical value of $\epsilon$ the stationary front loses stability and bifurcates into a left and a right moving wave. (b) If $2\kappa (1+\beta )>1$ then there is a single left-moving wave for all $\epsilon$ and a pair of right-moving waves that annihilate in a saddle-node bifurcation. Left and right moving waves are reversed when $2\kappa (1+\beta )<1$ (not shown).Reuse & Permissions

Figure 2

Stability phase diagram for a stationary front in the case of a step input $I(x)=-s\tanh (\gamma x)/2$ where $\gamma$ is the steepness of the step and $s$ its height. Hopf bifurcation lines (solid curves) in $s-\beta$ parameter space are shown for various values of $\epsilon$. In each case the stationary front is stable above the line and unstable below it. The shaded area denotes the region of parameter space where a stationary front solution does not exist. The threshold $\kappa =0.25$ and $\gamma =0.5$.Reuse & Permissions

Figure 3

Breatherlike solution arising from a Hopf instability of a stationary front due to a slow reduction in the size $s$ of a step input inhomogeneity and exponential weights. Here $\epsilon =0.5$, $\gamma =0.5$, $\beta =1$, and $\kappa =0.25$. The input amplitude $s=2$ at $t=0$ and $s=0$ at $t=180$. The amplitude of the oscillation steadily grows until it destabilizes at $s\approx 0.05$, leading to the generation of a traveling front.Reuse & Permissions

Figure 4

Plot of pulse width $a$ as a function of input amplitude $\mathcal{I}$ obtained by numerically solving Eq. (11) for $\sigma =1$, $\kappa =0.5$, and $\beta =1$. The lower branch is unstable whereas the upper branch is stable for large pulse width. (a) If $\epsilon >\beta$ then the upper branch undergoes a saddle-node bifurcation at $\mathcal{I}={\mathcal{I}}_{\mathrm{S}\mathrm{N}}$. (b) If $\epsilon <\beta$ then the upper branch undergoes a Hopf bifurcation at $\mathcal{I}={\mathcal{I}}_{\mathrm{H}\mathrm{B}}>{\mathcal{I}}_{\mathrm{S}\mathrm{N}}$.Reuse & Permissions

Figure 5

Breatherlike solution arising from a Hopf instability of a stationary pulse due to a slow reduction in the amplitude $\mathcal{I}$ of a Gaussian input and exponential weights. Here $\mathcal{I}=5.5$ at $t=0$ and $\mathcal{I}=1.5$ at $t=250$. Other parameter values are $\epsilon =0.03$, $\beta =2.5$, $\kappa =0.3$, and $\sigma =1.0$. The amplitude of the oscillation steadily grows until it undergoes a secondary instability at $\mathcal{I}\approx 2$, beyond which the breather persists and periodically generates pairs of traveling pulses (only one of which is shown). The breather itself disappears when $\mathcal{I}\approx 1$.Reuse & Permissions