×

Stationary periodic patterns in the 1D Gray-Scott model. (English) Zbl 0996.92041

Summary: We study the existence and stability of a family of stationary periodic patterns in the 1D Gray-Scott model. First, it is shown that these periodic solutions are born at a critical parameter value in a Turing/Ginzburg-Landau bifurcation, and an analysis of the appropriate Ginzburg-Landau normal form equation reveals that they exist below the critical parameter. Next, we analytically continue this family of periodic solutions in the ordinary differential equation for stationary solutions from the regime in which they are born and in which their spatial periods are \({\mathcal O}(1)\), to the regime where their spatial periods are asymptotically large. Depending on parameter values, the family terminates in global bifurcations via homoclinic orbits or in local bifurcations.
In addition to establishing these existence results, we perform a stability analysis. For parameter values near the critical parameter, there is an Eckhaus subband of stable periodic states within each existence interval. Moreover, these subbands of stable periodic states are continued into the full parameter range in which existence is shown. This numerical continuation is carried out all the way down into the regime where the periods of the orbits are asymptotically large and for which we have recently published analytical stability results. Taken together, these stability results show that there is a Busse balloon of stable stationary periodic solutions in the parameter space.
Finally, in numerical simulations, these stable periodic states are observed to be attractors for a wide variety of initial data, including data consisting of large-amplitude fronts moving into intervals over which the concentrations are in a linearly stable homogeneous state, data consisting of small-amplitude (Swift-Hohenberg like) fronts moving into intervals on which the concentrations are in a linearly unstable homogeneous state, and general oscillatory data.

MSC:

92E20 Classical flows, reactions, etc. in chemistry
34C25 Periodic solutions to ordinary differential equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
35K57 Reaction-diffusion equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N99 Applications of dynamical systems
34D35 Stability of manifolds of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI