28 research outputs found
A geometric analysis of the Lagerstrom model problem
AbstractLagerstrom's model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low Reynolds number by the method of matched asymptotic expansions. In this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems, in particular invariant manifold theory. As an essential part of the dynamics takes place near a line of non-hyperbolic equilibria, a blow-up transformation is introduced to resolve these singularities. This approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem. In particular, the source of the logarithmic switchback phenomenon is identified
Long-time behaviour of a model for p62-ubiquitin aggregation in cellular autophagy
The qualitative behavior of a recently formulated ODE model for the dynamics
of heterogenous aggregates is analyzed. Aggregates contain two types of
particles, oligomers and cross-linkers. The motivation is a preparatory step of
cellular autophagy, the aggregation of oligomers of the protein p62 in the
presence of ubiquitin cross-linkers. A combination of explicit computations,
formal asymptotics, and numerical simulations has led to conjectures on the
bifurcation behavior, certain aspects of which are proven rigorously in this
work. In particular, the stability of the zero state, where the model has a
smoothness deficit is analyzed by a combination of regularizing transformations
and blow-up techniques. On the other hand, in a different parameter regime, the
existence of polynomially growing solutions is shown by Poincar\'e
compactification, combined with a singular perturbation analysis
Geometric analysis of oscillations in the Frzilator model
A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it
Canards in a bottleneck
In this paper we investigate the stationary profiles of a nonlinear
Fokker-Planck equation with small diffusion and nonlinear in- and outflow
boundary conditions. We consider corridors with a bottleneck whose width has a
global nondegenerate minimum in the interior. In the small diffusion limit the
profiles are obtained constructively by using methods from geometric singular
perturbation theory (GSPT). We identify three main types of profiles
corresponding to: (i) high density in the domain and a boundary layer at the
entrance, (ii) low density in the domain and a boundary layer at the exit, and
(iii) transitions from high density to low density inside the bottleneck with
boundary layers at the entrance and exit. Interestingly, solutions of the last
type involve canard solutions generated at the narrowest point of the
bottleneck. We obtain a detailed bifurcation diagram of these solutions in
terms of the in- and outflow rates. The analytic results based on GSPT are
further corroborated by computational experiments investigating corridors with
bottlenecks of variable width.Comment: arXiv admin note: text overlap with arXiv:2010.1442
Fast and slow waves in the FitzHugh-Nagumo equation
It is known that the FitzHugh-Nagumo equation possesses fast and slow travelling waves. Fast waves are perturbations of singular orbits consisting of two pieces of slow manifolds and connections between them, whereas slow waves are perturbations of homoclinic orbits of the unperturbed system. We unfold a degenerate point where the two types of singular orbits coalesce forming a heteroclinic orbit of the unpertubed system. Let c denote the wave speed and ∈ the singular perturbation parameter. We show that there exists a C2 smooth curve of homoclinic orbits of the form (c,∈(c)) connecting the fast wave branch to the slow wave branch. Additionally we show that this curve has a unique non-degenerate maximum. Our analysis is based on a Shilnikov coordinates result, extending the Exchange Lemma of Jones and Kopell. We also prove the existence of inclination-filp points for the travelling wave equation thus providing the evidence of the existence of n-homoclinic orbits (n-pulses for the FitzHugh-Nagumo equation) for arbitrary n
A Skin Microbiome Model with AMP interactions and Analysis of Quasi-Stability vs Stability in Population Dynamics
The skin microbiome plays an important role in the maintenance of a healthy
skin. It is an ecosystem, composed of several species, competing for resources
and interacting with the skin cells. Imbalance in the cutaneous microbiome,
also called dysbiosis, has been correlated with several skin conditions,
including acne and atopic dermatitis. Generally, dysbiosis is linked to
colonization of the skin by a population of opportunistic pathogenic bacteria.
Treatments consisting in non-specific elimination of cutaneous microflora have
shown conflicting results. In this article, we introduce a mathematical model
based on ordinary differential equations, with 2 types of bacteria populations
(skin commensals and opportunistic pathogens) and including the production of
antimicrobial peptides to study the mechanisms driving the dominance of one
population over the other. By using published experimental data, assumed to
correspond to the observation of stable states in our model, we reduce the
number of parameters of the model from 13 to 5. We then use a formal
specification in quantitative temporal logic to calibrate our model by global
parameter optimization and perform sensitivity analyses. On the time scale of 2
days of the experiments, the model predicts that certain changes of the
environment, like the elevation of skin surface pH, create favorable conditions
for the emergence and colonization of the skin by the opportunistic pathogen
population, while the production of human AMPs has non-linear effect on the
balance between pathogens and commensals. Surprisingly, simulations on longer
time scales reveal that the equilibrium reached around 2 days can in fact be a
quasi-stable state followed by the reaching of a reversed stable state after 12
days or more. We analyse the conditions of quasi-stability observed in this
model using tropical algebraic methods, and show their non-generic character in
contrast to slow-fast systems. These conditions are then generalized to a large
class of population dynamics models over any number of species.Comment: arXiv admin note: substantial text overlap with arXiv:2206.1022
Singular perturbation analysis of a regularized MEMS model
Micro-Electro Mechanical Systems (MEMS) are defined as very small structures
that combine electrical and mechanical components on a common substrate. Here,
the electrostatic-elastic case is considered, where an elastic membrane is
allowed to deflect above a ground plate under the action of an electric
potential, whose strength is proportional to a parameter . Such
devices are commonly described by a parabolic partial differential equation
that contains a singular nonlinear source term. The singularity in that term
corresponds to the so-called "touchdown" phenomenon, where the membrane
establishes contact with the ground plate. Touchdown is known to imply the
non-existence of steady state solutions and blow-up of solutions in finite
time. We study a recently proposed extension of that canonical model, where
such singularities are avoided due to the introduction of a regularizing term
involving a small "regularization" parameter . Methods from
dynamical systems and geometric singular perturbation theory, in particular the
desingularization technique known as "blow-up", allow for a precise description
of steady-state solutions of the regularized model, as well as for a detailed
resolution of the resulting bifurcation diagram. The interplay between the two
main model parameters and is emphasized; in particular,
the focus is on the singular limit as both parameters tend to zero