Search results for "Singular perturbation"
showing 10 items of 11 documents
Dynamical Features of the MAP Kinase Cascade
2017
The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focuses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the application…
Sustained oscillations in the MAP kinase cascade.
2016
Abstract The MAP kinase cascade is a network of enzymatic reactions arranged in layers. In each layer occurs a multiple futile cycle of phosphorylations. The fully phosphorylated substrate then serves as an enzyme for the layer below. This paper focuses on the existence of parameters for which Hopf bifurcations occur and generate periodic orbits. Furthermore it is explained how geometric singular perturbation theory allows to generalize results from simple models to more complex ones.
Convergence of iterative methods in perturbation theory
1995
We discuss iterative KAM type methods for eigenvalue problems in finite dimensions. We compare their convergence properties with those of straight forward power series expansions.
Comparison of parallel implementation of some multi-level Schwarz methods for singularly perturbed parabolic problems
1999
Abstract Parallel multi-level algorithms combining a time discretization and an overlapping domain decomposition technique are applied to the numerical solution of singularly perturbed parabolic problems. Two methods based on the Schwarz alternating procedure are considered: a two-level method with auxiliary “correcting” subproblems as well as a three-level method with auxiliary “predicting” and “correcting” subproblems. Moreover, modifications of the methods using time extrapolation on subdomain interfaces are investigated. The emphasis is given to the description of the algorithms as well as their computer realization on a distributed memory multiprocessor computer. Numerical experiments …
Strong-coupling expansions for the -symmetric oscillators
1998
We study the traditional problem of convergence of perturbation expansions when the hermiticity of the Hamiltonian is relaxed to a weaker symmetry. An elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models. We describe its perturbative features paying particular attention to the strong-coupling regime. Efficient numerical perturbation theory proves suitable for such a purpose.
A proof of bistability for the dual futile cycle
2014
Abstract The multiple futile cycle is an important building block in networks of chemical reactions arising in molecular biology. A typical process which it describes is the addition of n phosphate groups to a protein. It can be modelled by a system of ordinary differential equations depending on parameters. The special case n = 2 is called the dual futile cycle. The main result of this paper is a proof that there are parameter values for which the system of ODE describing the dual futile cycle has two distinct stable stationary solutions. The proof is based on bifurcation theory and geometric singular perturbation theory. An important entity built of three coupled multiple futile cycles is…
Smooth Feshbach map and operator-theoretic renormalization group methods
2003
Abstract A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map . It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.
Simplification of Models
2016
In practical applications the “complete” model, i.e., a model that contains all features that the experts in the application domain consider important, is often quite complicated and difficult to analyse mathematically. A straightforward numerical realization is often costly and may give very little qualitative understanding of the situation. It is therefore important to study if the model can be systematically simplified in order to enhance a qualitative analysis/understanding.
Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity
2001
Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.
Multiple Canard Cycles in Generalized Liénard Equations
2001
AbstractThe paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in “Canard Cycles and Center Manifolds” (F. Dumortier and R. Roussarie, 1996, Mem. Amer. Math. Soc., 121). The presentation is limited to generalized Liénard equations εx+α(x, c)x+β(x, c)=0.