In this paper, we consider the existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field. We prove the existence of three types of singular steady-state patterns (double fronts, single spikes, and double spikes) by solving their respective consistency conditions. In the case of single spikes, we prove the stability of single small spike solution for sufficiently large spatial period by studying an explicit non-local eigenvalue problem which is equivalent to the original eigenvalue problem. For the other solutions, we prove the instability by using the variational characterisation of eigenvalues. Finally, we present the results of some numerical computations of spike solutions based on the finite difference methods of Crank–Nicolson and Adams–Bashforth.

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.

A spike solution is constructed on the infinite line for a sub-diffusive version of the Gierer–Meinhardt reaction – diffusion model. A non-local eigenvalue problem governs the spike's stability and is explicitly solvable for a certain choice of the kinetic parameters. Its solution generalises former results for the Gierer–Meinhardt model with regular diffusion, and the normal and anomalous systems' properties are juxtaposed. It is shown that a Hopf bifurcation occurs in the sub-diffusive system for larger values of the time parameter τo as compared to the normal counterpart, rendering the anomalous system more stable. Asymptotic solutions are obtained near important values of the diffusion anomaly index γ and collectively shown to be valid over most of the applicable range 0 < γ < 1. A bifurcation delay scenario is described for the sub-diffusive system, and the WKB exponent is computed analytically.