arXiv subjects

Valery G. Romanovski

Publications and source records attributed to Valery G. Romanovski.

At least 19 recordsLinked to original sources

arXiv2026

Poincarè-Dulac Normal Forms: Formal and Analytic Aspects

We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincarè-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↗

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

First integrals and invariants of systems of ODEs

We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincaré-Dulac normal forms for autonomous systems of ODEs with diagonal matrix of the linear part. Using tools from computational algebra, we develop an algorithmic approach for identifying generators of the algebras of monomial and polynomial first integrals, which works in the general case where the matrix of the linear part includes algebraic complex eigenvalues. Our method also provides a practical tool for exploring the algebraic structure of polynomial invariants and their relation to the Poincaré-Dulac normal forms of the underlying vector fields.

math.DS↗

arXiv2024

Invariants and reversibility in polynomial systems of ODEs

This paper explores a relationship between invariants of certain group actions and the time-reversibility of two-dimensional polynomial differential systems exhibiting a $1:-1$ resonant singularity at the origin. We focus on the connection of time-reversibility with the Sibirsky subvariety of the center (integrability) variety, which encompasses systems possessing a local analytic first integral near the origin. An algorithm for generating the Sibirsky ideal for these systems is proposed and the algebraic properties of the ideal are examined. Furthermore, using a generalization of the concept of time-reversibility we study $n$-dimensional systems with a $1:ζ:ζ^2:\dots:ζ^{n-1}$ resonant singularity at the origin, where $n$ is prime and $ζ$ is a primitive $n$-th root of unity. We study the invariants of a Lie group action on the parameter space of the system, leveraging the theory of binomial ideals as a fundamental tool for the analysis. Our study reveals intriguing connections between generalized reversibility, invariants, and binomial ideals, shedding light on their complex interrelations.

math.DS↗

arXiv2023

Computation of Normal Forms for Systems with Many Parameters

There are two ways to compute Poincaré-Dulac normal forms of systems of ODEs. Under the original approach used by Poincaré the normalizing transformation is explicitly computed. On each step, the normalizing procedure requires the substitution of a polynomial to a series. Under the other approach, a normal form is computed using Lie transformations. In this case, the changes of coordinates are performed as actions of certain infinitesimal generators. In both cases, on each step the homological equation is solved in the vector space of polynomial vector fields $V^n_j$ where each component of the vector field is a homogeneous polynomial of degree $j$. We present the third way of computing normal forms of polynomial systems of ODEs where the coefficients of all terms are parameters. Although we use Lie transforms the homological equation is solved not in $V^n_j$ but in the vector space of polynomial vector fields where each component is a homogeneous polynomial in the parameters of the system. It is shown that the space of the parameters is a kind of dual space and, the computation of normal forms can be performed in the space of parameters treated as the space of generalized vector fields. The approach provides a simple way to parallelize the normal form computations opening the way to compute normal forms up to higher order than under previously known two approaches.

math.DS↗

arXiv2022

Time-reversibility and integrability of p:-q resonant vector fields

We study local analytical integrability in a neighborhood of $p:-q$ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ φ(x,y)=(y^{p/q},x^{q/p}). $ Some generalizations of the theory developed by K.~S.~Sibirsky for $1:-1$ resonant case to the $p:-q$ resonant case are presented.

math.DS↗

arXiv2020

Invariant Tori of a Two Dimensional Periodic System with the Linear-Cubic Unperturbed Part

Two classes of time-periodic systems of ordinary differential equations with a small nonnegative parameter, those with fast and slow time, are studied. Right-hand sides of these systems are three times continuously differentiable with respect to phase variables and the parameter, the corresponding unperturbed systems are autonomous, conservative and have nine equilibrium points. For the perturbed systems, which do not depend on the parameter explicitly, we obtain the conditions yielding that the initial system has a certain number of two-dimensional invariant surfaces homeomorphic to a torus for each sufficiently small values of parameter and the formulas of such surfaces. A class of systems with seven invariant surfaces enclosing different configurations of equilibrium points is studied as an example of applications of our method.

math.DS↗

arXiv2018

On three genetic repressilator topologies

Novel mathematical models of three different repressilator topologies are introduced. As designable transcription factors have been shown to bind to DNA non-cooperatively, we have chosen models containing non-cooperative elements. The extended topologies involve three additional transcription regulatory elements---which can be easily implemented by synthetic biology---forming positive feedback loops. This increases the number of variables to six, and extends the complexity of the equations in the model. To perform our analysis we had to use combinations of modern symbolic algorithms of computer algebra systems Mathematica and Singular. The study shows that all the three models have simple dynamics what can also be called regular behaviour: they have a single asymptotically stable steady state with small amplitude oscillations in the 3D case and no oscillation in one of the 6D cases and damping oscillation in the second 6D case. Using the program QeHopf we were able to exclude the presence of Hopf bifurcation in the 3D system.

math.DS↗

arXiv2017

Linearizability and critical period bifurcations of a generalized Riccati system

In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

math.DS↗

arXiv2017

Centers and limit cycles of a generalized cubic Riccati system

We obtain condition for existence of a center for a cubic planar differential system, which can be considered as a polynomial subfamily of the generalized Riccati system. We also investigate bifurcations of small limit cycles from the components of the center variety of the system.

math.DS↗

arXiv2016

Isochronicity and linearizability of a planar cubic system

In this paper we investigate the problem of linearizability for a family of cubic complex planar systems of ordinary differential equations. We give a classification of linearizable systems in the family obtaining conditions for linearizability in terms of parameters. We also discuss coexistence of isochronous centers in the systems.

math.DS↗

arXiv2014

Groebner basis methods for stationary solutions of a low-dimensional model for a shear flow

We use Groebner basis methods to extract all stationary solutions for the 9-mode shear flow model that is described in Moehlis et al, New J. Phys. 6, 54 (2004). Using rational approximations to irrational wave numbers and algebraic manipulation techniques we reduce the problem of determining all stationary states to finding roots of a polynomial of order 30. The coefficients differ by 30 powers of 10 so that algorithms for extended precision are needed to extract the roots reliably. We find that there are eight stationary solutions consisting of two distinct states that each appear in four symmetry-related phases. We discuss extensions of these results for other flows.

physics.flu-dyn↗

arXiv2012

Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials

In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. Based on the method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be elementary center or nilpotent center. Under the condition for the singular point to be a center, we obtain the normal form of the Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating from the center using the algorithm to compute the coefficients of Melnikov function. Finally, perturbing the symmetric hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is same to that of another center.

math.CA↗

arXiv2011

Bifurcations of planar Hamiltonian systems with impulsive perturbation

In this paper, by means of the Melnikov functions we consider bifurcations of harmonic or subharmonic solutions from a periodic solution of a planar Hamiltonian system under impulsive perturbation. We give some sufficient conditions under which a harmonic or subharmonic solution exists.

math.CA↗

arXiv2011

On the number of limit cycles of polynomial Lienard systems

Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.

math.CA↗

arXiv2011

Limit cycle bifurcations from a nilpotent focus or center of planar systems

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center.

math.CA↗

arXiv2011

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclnic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.

math.CA↗

arXiv2007

Time-Reversibility in 2-Dim Systems

We present an algorithm for finding all time-reversible systems within a given family of 2-dim systems of ODE's whose right-hand sides are polynomials. We also study an interconnection of time-reversibility and invariants of a subgroup of SL(2,C).

math.DS↗