Xingwu Chen

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 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.

Decomposition of algebraic sets and applications to weak centers of cubic systems

There are many methods such as Grobner basis, characteristic set and resultant, in computing an algebraic set of a system of multivariate polynomials. The common difficulties come from the complexity of computation, singularity of the corresponding matrices and some unnecessary factors in successive computation. In this paper, we decompose algebraic sets, stratum by stratum, into a union of constructible sets with Sylvester resultants, so as to simplify the procedure of elimination. Applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center, we determine the order of weak centers and discuss the bifurcation of critical periods.

Isochronicity of analytic systems via Urabe's criterion

A method for studying isochronous oscillations in some systems of ODE reducible to the equation is described. It is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two-dimensional autonomous system depending on six parameters. For all isochronous systems in this family the Urabe function is explicitly constructed.

Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system

Since Chicone and Jacobs investigated local bifurcation of critical periods for quadratic systems and Newtonian systems in 1989, great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation. This paper is devoted to the linearizability and local bifurcation of critical periods for a cubic Kolmogorov system. We use the Darboux method to give explicit linearizing transformations for isochronous centers. Investigating the finite generation for the ideal of all period constants, which are of the polynomial form in six parameters, we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center. Moreover, we prove that the maximum number of critical periods is reachable.

Local bifurcations of critical periods in a generalized 2D LV system

Abstract In this paper, we investigate a generalized two-dimensional Lotka–Volterra system which has a center. We give an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order. We show that with appropriate perturbations at most two critical periods bifurcate from the center.

Isochronicity of centers in some planar differential systems

For planar differential systems the isochronicity of centers, as a continuation of the center problem, relates to the synchronism of periodic oscillations. In this paper we introduce recent results and basic methods on isochronicity of nondegenerate centers for planar differential systems including homogeneous systems, reversible systems, and Hamiltonian systems.

LINEAR CENTERS WITH PERTURBATIONS OF DEGREE 2d + 5

We describe a method for studying the center and isochronicity problems for a class of differential systems in the form of linear center perturbed by homogeneous series of degree 2d + m, where d is a non-negative real number and m is a positive integer. As an application, we classify centers and isochronous centers for a particular case when m = 5.