Brigita Ferčec

Investigation of center manifolds of three-dimensional systems using computer algebra

For a family of three dimensional systems with center manifolds filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well. The study is performed using computer algebra systems MATHEMATICA and SINGULAR.

Original article: Center conditions and cyclicity for a family of cubic systems: Computer algebra approach

Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. We overcame the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we determine the number of limit cycles bifurcating from each component of the center variety.

The solution of the 1: 3 resonant center problem in the quadratic case

Abstract The 1:−3 resonant center problem in the quadratic case is to find necessary and sufficient conditions for the existence of a local analytic first integral for the differential system x = x - a 10 x 2 - a 01 xy - a 12 y 2 , y = - 3 y + b 21 x 2 + b 10 xy + b 01 y 2 . There appear 25 center cases for a 10 = 1 and 11 cases for a 01 = 0 . The necessity is obtained using modular arithmetics and with high probability we have all the resonant center cases. We show that in each case there exists a local analytic first integral (sufficient condition) around the origin. This sufficient condition is proved using different classical criteria and in one case monodromy arguments.

Cyclicity of some analytic maps

Abstract In this paper, we describe an approach to estimate the cyclicity of centers in maps given by f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 . The main motivation for this problem originates from the study of cyclicity of planar systems of ODEs. We also consider the bifurcation of limit cycles from each component of the center variety of some particular cases of maps f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 arising from algebraic equations of the form x + y + h . o . t . = 0 where higher order terms up to degree four are present.

On integrability conditions and limit cycle bifurcations for polynomial systems

We present an approach to finding general conditions for integrability of a given family of two-dimensional polynomial systems using conditions computed when some parameters were fixed. We apply it to obtain integrability conditions for a Lotka-Volterra planar complex quartic system having homogeneous nonlinearities. We also study bifurcations of limit cycles from each component of the center variety of the corresponding quartic real system.

Isochronicity of centers at a center manifold

For a three dimensional system with a center manifold filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well.

Dynamical analysis of a two prey-one predator system with quadratic self interaction

Abstract In this paper we investigate the dynamical properties of a two prey-one predator system with quadratic self interaction represented by a three-dimensional system of differential equations by using tools of computer algebra. We first investigate the stability of the singular points. We show that the trajectories of the solutions approach to stable singular points under given conditions by numerical simulation. Then, we determine the conditions for the existence of the invariant algebraic surfaces of the system and we give the invariant algebraic surfaces to study the flow on the algebraic invariants which is a useful approach to check if Hopf bifurcation exists.

The center and cyclicity problems for some analytic maps

Abstract The center variety and bifurcations of limit cycles from the center for maps f ( x ) = − ∑ k = 0 ∞ a k x k + 1 arising from x + y + ∑ j = 0 n α n − j , j x n − j y j = 0 are considered. Motivated by a general result for n = 2 l + 1 we investigate the center and cyclicity problem for n being even. We review results for n = 2 and n = 4 and perform the analysis for n = 6 , 8 , 10 . Finally, we state some conjectures for general n = 2 l .