Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model

Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic Hopf frequency. Such solutions of this nonvariational problem are in contrast to the stationary pinning solutions found in the subcritical Turing regime for the variational Swift-Hohenberg equations, characterized by a spatially periodic pattern embedded in a spatially homogeneous background state. Numerical continuation was used to solve periodic boundary value problems in time for the Fourier amplitudes of the spatiotemporal Turing-Hopf pinning solutions. The solution branches are organized in a series of saddle-node bifurcations similar to the known snaking structures of stationary pinning solutions. We find two intertwined pairs of such branches, one with a defect in the middle of the striped region, and one without. Solutions on one branch of one pair differ from those on the other branch by a π phase shift in the spatially periodic region, i.e., locations of local minima of solutions on one branch correspond to locations of maxima of solutions on the other branch. These branches are connected to branches exhibiting collapsed snaking behavior, where the snaking region collapses to almost a single value in the bifurcation parameter. Solutions along various parts of the branches are described in detail. Time dependent depinning dynamics outside the saddle nodes are illustrated, and a time scale for the depinning transitions is numerically established. Wavelength variation within the snaking region is discussed, and reasons for the variation are given in the context of amplitude equations. Finally, we compare the pinning region to the Maxwell line found numerically by time evolving the amplitude equations.