arXiv subjects

Sebastian Walcher

Publications and source records attributed to Sebastian Walcher.

At least 19 recordsLinked to original sources

arXiv2026

Lumping of reaction networks: Generic and critical parameters

We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters -- those ranging in some non-empty open subset of parameter space -- we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets -- algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values

math.DS↗

arXiv2025

Liouvillian integrability of vector fields in higher dimensions

We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.

nlin.SI↗

arXiv2025

Liouvillian integrability of rational vector fields: The case of algebraic extensions

As shown in a previous paper, whenever a rational vector field on $\mathbb C^n$, $n>2$, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite algebraic extension $L$ of the rational function field $K$. In the present work we discuss and characterize exceptional vector fields in this class, for which -- by definition -- the choice $L=K$ is not possible. In particular we show that exceptional vector field exist, giving explicit constructions in dimension three.

math.RA↗

arXiv2025

Poincar\`e-Dulac Normal Forms: Formal and Analytic Aspects

We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincar\`e-Dulac normal form, and the convergence of such transformations. After introducing a new transparent approach to transformations, we review Bruno's approach to formal normalization, as well as convergence results in presence of certain versions of Bruno's Condition A. Retracing the proof steps in Bruno's work, we use a different formalism and variants in the line of arguments, which are easily transferrable to more general scenarios. In particular we show how Bruno's approach naturally extends to an elementary proof of Stolovitch's formal and analytic simultaneous normalization theorems for abelian Lie algebras of vector fields. Furthermore we investigate the role of (meromorphic and formally meromorphic) integrability in normal forms and convergence properties. In particular we establish a bridge to Zung's convergence theorems and show that these are can be complemented by Stolovitch's results.

math.DS↗

arXiv2025

Resnikoff silver numbers and tilings of the half-line (Dedicated to the memory of H.L.Resnikoff)

Building on work by H.L.Resnikoff we consider (Resnikoff) silver numbers, which generalize the familiar golden number. By definition, a silver number is the largest positive root of a certain polynomial called silver polynomial. In turn, a corresponding companion matrix of a silver polynomial gives rise to a well known construction of inflationary tilings of the (non-negative) real half-line, via an iteration of inflation and substitution. Resnikoff noted for the golden number $\phi$ that this tiling corresponds to the set of what he called $\phi$-integers. We generalize this result for a special class of silver numbers, the distinguished silver numbers, by showing that the integers for a distinguished silver number give rise to a tiling, of which we provide a precise description. For the general problem, whether the integers for an arbitrary silver number give rise to a tiling, we cannot give a general answer, but we show that tilings are obtained if and only if the differences of silver integers satisfy a (rather weak looking) non-accumulation condition. If tilings of this type exist for certain (necessarily non-distinguished) silver numbers, they would seem to form a class of inflationary tilings that differs from those obtained by inflation and substitution. In an Appendix we recall necessary notions and -- mostly known -- results, including the inflation-substitution construction principle for (one dimensional) inflationary tilings, in an elementary manner. For the readers' convenience we also collect the pertinent facts about non-negative matrices, thus the construction is accessible with only basic prerequisites from linear algebra and analysis. Finally, in our setting we give a detailed proof of a non-periodicity result that goes back to Penrose.

math.CO↗

arXiv2025

Floquet-like theory and gauge transformations for general smooth dynamical systems

The classical Floquet theory allows to map a time-periodic system of linear differential equations into an autonomous one. By looking at it in a geometrical way, we extend the theory to a class of non-autonomous non-periodic equations. This is obtained by considering a change of variables which depends on time in a non-trivial way, i.e. introducing gauge transformations, well known in fundamental Physics and Field Theory -- but which seems to have received little attention in this context.

math-ph↗

arXiv2025

On gauge transforms of autonomous ordinary differential equations

The notion of gauge transform has its origin in Physics (Field Theory). In the present note we discuss -- from a purely mathematical perspective -- special gauge transforms of autonomous first order ODE's and their special properties. Particular attention is given to the problem of identifying those nonautonomous ODE's which are gauge transforms of autonomous systems.

math.DS↗

arXiv2024

On elementary integrability of rational vector fields

We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field $K$. In view of the Prelle-Singer theorem, these are the rational vector fields that admit an elementary first integral. Elementary integrable vector fields which are not Darboux integrable -- thus the extension field is necessarily a proper extension of $K$ -- may be called exceptional by an observation in an earlier paper by Christopher et al. For dimension two we characterize all possible algebraic extension fields underlying the exceptional cases, provide a construction of all exceptional vector fields, and obtain some criteria that restrict the degree of $L$.

math.DS↗

arXiv2024

The Michaelis--Menten reaction at low substrate concentrations: Pseudo-first-order kinetics and conditions for timescale separation

We demonstrate that the Michaelis-Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke's comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is $s_0 \ll K$, where $K=k_2/k_1$ is the Van Slyke-Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

math.DS↗

arXiv2023

Liouvillian integrability of three dimensional vector fields

We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability {in any dimension $n\geq 3$} always implies the existence of a first integral that is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field.

nlin.SI↗

arXiv2023

Rigorous estimates for the quasi-steady state approximation of the Michaelis-Menten reaction mechanism at low enzyme concentrations

There is a vast amount of literature concerning the appropriateness of various perturbation parameters for the standard quasi-steady state approximation in the Michaelis-Menten reaction mechanism, and also concerning the relevance of these parameters for the accuracy of the approximation by the familiar Michaelis-Menten equation. Typically, the arguments in the literature are based on (heuristic) timescale estimates, from which one cannot obtain reliable quantitative estimates for the error of the quasi-steady state approximation. We take a different approach. By combining phase plane analysis with differential inequalities, we derive sharp explicit upper and lower estimates for the duration of the initial transient and substrate depletion during this transitory phase. In addition, we obtain rigorous bounds on the accuracy of the standard quasi-steady state approximation in the slow dynamics regime. Notably, under the assumption that the quasi-steady state approximation is valid over the entire time course of the reaction, our error estimate is of order one in the Segel-Slemrod parameter.

math.DS↗

arXiv2022

Natural parameter conditions for singular perturbations of chemical and biochemical reaction networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis--Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

math.DS↗

arXiv2021

On the anti-quasi-steady-state conditions of enzyme kinetics

Quasi-steady state reductions for the irreversible Michaelis--Menten reaction mechanism are of interest both from a theoretical and an experimental design perspective. A number of publications have been devoted to extending the parameter range where reduction is possible, via improved sufficient conditions. In the present note, we complement these results by exhibiting local conditions that preclude quasi-steady-state reductions (anti-quasi-steady-state), in the classical as well as in a broader sense. To this end, one needs to obtain necessary (as opposed to sufficient) conditions and determine parameter regions where these do not hold. In particular, we explicitly describe parameter regions where no quasi-steady-state reduction (in any sense) is applicable (anti-quasi-steady-state conditions), and we also show that -- in a well defined sense -- these parameter regions are small. From another perspective, we obtain local conditions for the accuracy of standard or total quasi-steady-state. Perhaps surprisingly, our conditions do not involve initial substrate.

physics.chem-ph↗

arXiv2021

Critical parameters for singular perturbation reductions of chemical reaction networks

We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path toward such reductions. In the present paper we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov-Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero), or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex balanced reaction networks (of deficiency zero) and first order reaction networks.

math.DS↗

arXiv2021

On the quasi-steady-state approximation in an open Michaelis--Menten reaction mechanism

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis--Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant slow manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.

math.DS↗

arXiv2020

Algorithmic Reduction of Biological Networks With Multiple Time Scales

We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

q-bio.MN↗

arXiv2020

Higher order normal modes

Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.

math.DS↗

arXiv2020

Centralizers and normalizers of local analytic and formal vector fields

We investigate the structure of the centralizer and the normalizer of a local analytic or formal differential system at a nondegenerate stationary point, using the theory of Poincar\'e-Dulac normal forms. Our main results are concerned with the formal case. We obtain a description of the relation between centralizer and normalizer, sharp dimension estimates when the centralizer of the linearization has finite dimension, and lower estimates for the dimension of the centralizer in general. For a distinguished class of linear vector fields (which is sufficiently large to be of interest) we obtain a precise characterization of the centralizer for corresponding normal forms in the generic case. Moreover, in view of their relation to normalizers, we discuss inverse Jacobi multipliers and obtain existence criteria and nonexistence results for several classes of vector fields.

math.DS↗