176 research outputs found
Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems
This article establishes the foundation for a new theory of
invariant/integral manifolds for non-autonomous dynamical systems. Current
rigorous support for dimensional reduction modelling of slow-fast systems is
limited by the rare events in stochastic systems that may cause escape, and
limited in many applications by the unbounded nature of PDE operators. To
circumvent such limitations, we initiate developing a backward theory of
invariant/integral manifolds that complements extant forward theory. Here, for
deterministic non-autonomous ODE systems, we construct a conjugacy with a
normal form system to establish the existence, emergence and exact construction
of center manifolds in a finite domain for systems `arbitrarily close' to that
specified. A benefit is that the constructed invariant manifolds are known to
be exact for systems `close' to the one specified, and hence the only error is
in determining how close over the domain of interest for any specific
application. Built on the base developed here, planned future research should
develop a theory for stochastic and/or PDE systems that is useful in a wide
range of modelling applications
Large dispersion, averaging and attractors: three 1D paradigms
The effect of rapid oscillations, related to large dispersion terms, on the
dynamics of dissipative evolution equations is studied for the model examples
of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three
different scenarios of this effect are demonstrated. According to the first
scenario, the dissipation mechanism is not affected and the diameter of the
global attractor remains uniformly bounded with respect to the very large
dispersion coefficient. However, the limit equation, as the dispersion
parameter tends to infinity, becomes a gradient system. Therefore, adding the
large dispersion term actually suppresses the non-trivial dynamics. According
to the second scenario, neither the dissipation mechanism, nor the dynamics are
essentially affected by the large dispersion and the limit dynamics remains
complicated (chaotic). Finally, it is demonstrated in the third scenario that
the dissipation mechanism is completely destroyed by the large dispersion, and
that the diameter of the global attractor grows together with the growth of the
dispersion parameter
Aggregation-diffusion equations: dynamics, asymptotics, and singular limits
Given a large ensemble of interacting particles, driven by nonlocal
interactions and localized repulsion, the mean-field limit leads to a class of
nonlocal, nonlinear partial differential equations known as
aggregation-diffusion equations. Over the past fifteen years,
aggregation-diffusion equations have become widespread in biological
applications and have also attracted significant mathematical interest, due to
their competing forces at different length scales. These competing forces lead
to rich dynamics, including symmetrization, stabilization, and metastability,
as well as sharp dichotomies separating well-posedness from finite time blowup.
In the present work, we review known analytical results for
aggregation-diffusion equations and consider singular limits of these
equations, including the slow diffusion limit, which leads to the constrained
aggregation equation, as well as localized aggregation and vanishing diffusion
limits, which lead to metastability behavior. We also review the range of
numerical methods available for simulating solutions, with special attention
devoted to recent advances in deterministic particle methods. We close by
applying such a method -- the blob method for diffusion -- to showcase key
properties of the dynamics of aggregation-diffusion equations and related
singular limits
Beginner's guide to Aggregation-Diffusion Equations
The aim of this survey is to serve as an introduction to the different
techniques available in the broad field of Aggregation-Diffusion Equations. We
aim to provide historical context, key literature, and main ideas in the field.
We start by discussing the modelling and famous particular cases: Heat
equation, Fokker-Plank, Porous medium, Keller-Segel,
Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez,
McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present
the well-posedness frameworks given as PDEs in Sobolev spaces, and
gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time,
for which we need to understand minimisers of a free energy. We then present
some numerical methods which have been developed. We conclude the paper
mentioning some related problems
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