The \({1:-q}\) resonant center problem for certain cubic Lotka-Volterra systems

DOI10.1016/j.amc.2012.05.045zbMath1365.34006OpenAlexW2055878696MaRDI QIDQ387701

Valery G. Romanovski, Douglas S. Shafer, Jaume Giné, Xingwu Chen

Publication date: 23 December 2013

Published in: Applied Mathematics and Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.amc.2012.05.045

zbMATH Keywords

first integralresonant center problempolynomial systems of differential equations

Mathematics Subject Classification ID

Periodic solutions to ordinary differential equations (34C25) Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Explicit solutions, first integrals of ordinary differential equations (34A05) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07)

Related Items (18)

The solution of the \(1:-3\) resonant center problem in the quadratic case ⋮ Analytic integrability of certain resonant saddle ⋮ Integrability of fractional order generalized systems with \(p:-q\) resonance ⋮ On the formal integrability problem for planar differential systems ⋮ Local integrability and linearizability of a \((1:-1:-1)\) resonant quadratic system ⋮ The center problem for a \({1:-4}\) resonant quadratic system ⋮ Integrability and linearizability of symmetric three-dimensional quadratic systems ⋮ The center problem for a \(2:-3\) resonant cubic Lotka-Volterra system ⋮ Analytical integrability of perturbations of quadratic systems ⋮ A BLOW-UP METHOD TO PROVE FORMAL INTEGRABILITY FOR SOME PLANAR DIFFERENTIAL SYSTEMS ⋮ The isochronous centers for Kukles homogeneous system of degree nine ⋮ Complex integrability and linearizability of cubic \(Z_2\)-equivariant systems with two \(1:q\) resonant singular points ⋮ The integrability of a class of cubic Lotka-Volterra systems ⋮ Analytically integrable centers of perturbations of cubic homogeneous systems ⋮ Integrability and linearizability for Lotka-Volterra systems with the \(3 : -q\) resonant saddle point ⋮ The complexity of generalized center problem ⋮ Integrability and generalized center problem of resonant singular point ⋮ Local integrability and linearizability of three-dimensional Lotka-Volterra systems

Uses Software

• FGb
• SINGULAR
• primdec

Cites Work

• Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials
• The Poincaré center problem
• Integrability of Lotka-Volterra type systems of degree 4
• On the integrability of two-dimensional flows
• The problem of center for resonant singular points of polynomial vector fields
• Normalizable, integrable, and linearizable saddle points for complex quadratic systems in \(\mathbb{C}^2\)
• Darboux integrability and the inverse integrating factor.
• On the number of algebraically independent Poincaré-Liapunov constants
• \(1: - 3\) resonant centers on \(\mathbb{C}^2\) with homogeneous cubic nonlinearities
• On the center problem for \(p:-q\) resonant polynomial vector fields
• The Center and Cyclicity Problems
• Probleme General de la Stabilite du Mouvement. (AM-17)
• Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in \(\mathbb C^2\)
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