Valery G. Romanovski

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.

Linearizability of linear systems perturbed by fifth degree homogeneous polynomials

We present the necessary and sufficient conditions for linearizability of the planar complex system , where P and Q are homogeneous polynomials of degree 5. Using these conditions, we also give the complete solution for the isochronicity of real systems in the form of linear oscillator perturbed by fifth degree homogeneous polynomials.

The centre and isochronicity problems for some cubic systems

We present an efficient method for computing focus and linearizability quantities of polynomial differential equation systems. We apply the method to computing these quantities for ten eight-parametric cubic systems and obtain the necessary and sufficient conditions of linearizability (isochronicity) of these systems. We also show that there is a kind of duality between the problem of constructing algebraic invariant curves, first integrals and linearizing transformations on one side, and the problem of solving some first-order linear partial differential equations on the other side.

The Sibirsky component of the center variety of polynomial differential systems

We investigate a component of the center variety of polynomial differential systems that includes all time-reversible systems. We give a general algorithm to find this irreducible subvariety and compute its dimension.

Linearizability conditions for Lotka?Volterra planar complex cubic systems

In this paper, we investigate the linearizability problem for the two-dimensional planar complex system . The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.

Time-Reversibility in 2-Dimensional Systems

We present an algorithm for finding the Zariski closure of the set of all time-reversible systems within a given family of two-dimensional autonomous systems of ODEs whose right-hand sides are polynomials. We also study an interconnection of time-reversibility and invariants of a subgroup of SL(2, ℂ).

The 1: −q resonant center problem for certain cubic Lotka–Volterra systems

Abstract Necessary conditions and distinct sufficient conditions are derived for the system x ˙ = x ( 1 - a 20 x 2 - a 11 xy - a 02 y 2 ) , y ˙ = y ( - q + b 20 x 2 + b 11 xy + b 02 y 2 ) to admit a first integral of the form Φ ( x , y ) = x q y + ⋯ in a neighborhood of the origin, in which case the origin is termed a 1 : - q resonant center. Necessary and sufficient conditions are obtained for odd q , q ⩽ 9 ; necessary conditions, most of which are also sufficient, are obtained for even q , q ⩽ 8 . Key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions, and the theory of Darboux first integrals to show sufficiency.

Critical period bifurcations of a cubic system

We study bifurcations of the period function of a linear centre perturbed by third degree homogeneous polynomials. The approach is based on making use of algorithms of computational algebra.

The resonant center problem for a 2: -3 resonant cubic lotka---volterra system

Using tools of computer algebra we derive the conditions for the cubic Lotka---Volterra system

Isochronicity of analytic systems via Urabe's criterion

\dot x = x( 2 - a_{20} x^2 - a_{11} xy - a_{02} y^2)