G. S. Medvedev and D. E. Pelinovsky
Turing bifurcation in the Swift-Hohenberg equation on deterministic and random graphs
Abstract:
The Swift-Hohenberg equation (SHE) is a partial differential equation that
explains how patterns emerge from a spatially homogeneous state. It has been widely used
in the theory of pattern formation. Following a recent study by Bramburger and Holzer,
we consider discrete SHE on deterministic and random graphs. The two families of
the discrete models share the same continuum limit in the form of a nonlocal SHE on a
circle. The analysis of the continuous system, parallel to the analysis of the classical SHE,
shows bifurcations of spatially periodic solutions at critical values of the control parameters.
However, the proximity of the discrete models to the continuum limit does not guarantee
that the same bifurcations take place in the discrete setting in general, because some of the
symmetries of the continuous model do not survive discretization.
We use the center manifold reduction and normal forms to obtain precise information
about the number and stability of solutions bifurcating from the homogeneous state in the
discrete models on deterministic and sparse random graphs. Moreover, we present detailed
numerical results for the discrete SHE on the nearest-neighbor and small-world graphs.
Keywords:
Swift-Hohenberg equation, discrete models, deterministic and random graphs, Turing bifurcation, center manifolds, normal forms.