Qualitative Theory of Dynamical Systems
Local Integrability and Linearizability for Three Dimensional Lotka–Volterra Cubic Systems
Abstract: In this study, we investigate the integrability and linearizability problems of a family of
cubic three-dimensional Lotka–Volterra systems with one zero eigenvalue, involving
seventeen parameters. Necessary conditions on the parameters of the system for both
integrability and linearizability are obtained by computing the resonant quantities
using Gröbner bases and decomposing the variety of the ideal generated in the ring
of polynomials of parameters of the system. The sufficiency of these conditions is
also proven except that for a case, Case 32, of sufficiency has been left as conjectural.
In particular, we used the Darboux method, the existence of a first integral with an
inverse Jacobi multiplier, time reversibility, the properties of linearizable nodes in two
dimensional systems and power series arguments to the third variable and some other
techniques
Subject: Integrability , Linearizability · , Invariant algebraic surface , Exponential factor , Inverse Jacobi multiplier , First integral
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| contributor author | Karim, Aween | |
| contributor author | Aziz, Waleed | |
| contributor author | Amen, Azad | |
| date accessioned | 2025-02-21T18:59:31Z | |
| date available | 2025-02-21T18:59:31Z | |
| date issued | 2025 | |
| identifier uri | http://192.64.112.23/xmlui/handle/311/89 | |
| description abstract | In this study, we investigate the integrability and linearizability problems of a family of cubic three-dimensional Lotka–Volterra systems with one zero eigenvalue, involving seventeen parameters. Necessary conditions on the parameters of the system for both integrability and linearizability are obtained by computing the resonant quantities using Gröbner bases and decomposing the variety of the ideal generated in the ring of polynomials of parameters of the system. The sufficiency of these conditions is also proven except that for a case, Case 32, of sufficiency has been left as conjectural. In particular, we used the Darboux method, the existence of a first integral with an inverse Jacobi multiplier, time reversibility, the properties of linearizable nodes in two dimensional systems and power series arguments to the third variable and some other techniques | en_US |
| language iso | en_US | en_US |
| publisher | Qualitative Theory of Dynamical Systems | en_US |
| subject | Integrability | en_US |
| subject | Linearizability · | en_US |
| subject | Invariant algebraic surface | en_US |
| subject | Exponential factor | en_US |
| subject | Inverse Jacobi multiplier | en_US |
| subject | First integral | en_US |
| title | Local Integrability and Linearizability for Three Dimensional Lotka–Volterra Cubic Systems | en_US |
| type | Article | en_US |