We consider the impact of spatial heterogeneities on the dynamics of
localized patterns in systems of partial differential equations (in one
spatial dimension). We will mostly focus on the most simple possible
heterogeneity: a small jump-like defect that appears in models in which
some parameters change in value as the spatial variable x crosses
through a critical value -- which can be due to natural inhomogeneities,
as is typically the case in ecological models, or can be imposed on the
model for engineering purposes, as in Josephson junctions. Even such a
small, simplified heterogeneity may have a crucial impact on the
dynamics of the PDE. We will especially consider the effect of the
heterogeneity on the existence of defect solutions, which boils down to
finding heteroclinic (or homoclinic) orbits in an n-dimensional
dynamical system in `time' x, for which the vector field for x > 0
differs slightly from that for x < 0 (under the assumption that there is
such an orbit in the homogeneous problem). Both the dimension of the
problem and the nature of the linearized system near the limit points
have a remarkably rich impact on the defect solutions. We complement the
general approach by considering two explicit examples: a heterogeneous
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous
generalized FitzHugh–Nagumo system (n = 6).
Seminar series
Date
Thu, 15 Oct 2015
Time
16:00 -
17:00
Location
L3
Speaker
Arjen Doelman
Organisation
Leiden University