Applied Maths Seminar: Pelinovsky -- Turing bifurcation on deterministic and random graphs

Dmitry Pelinovsky (McMaster University) is visiting us until 26th March.  He will be
giving a talk on **Wednesday** Week 1 at 11am in Carslaw 173.  

Note that Wednesday 11am is now the standard slot for the AM seminar.  

Afterwards we will go to lunch (which is free for students).  

Title: Turing Bifurcation 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 the 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.