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<p>The linearizability problem of a singular point at the origin for a family of a quadratic four dimensional Lotka–Volterra system is investigated for (1:−1:1:1)-resonance. A complete set of necessary and sufficient conditions is obtained on the parameters that guarantee the existence of a coordinate change which transforms the system to a linear system. Several methods and techniques are used to prove the sufficiency of the obtained conditions. Mainly the Darboux linearization method and the linearizability of a singular point at the origin in the Poincaré domain are used in three variables with power-series arguments in the fourth variable. In some cases, if the system is completely integrable and there is a substitution that linearizes one of the equations, then the system is linearzable.</p>

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Local first integrals of four dimensional Lotka–Volterra systems

  • 期刊论文

  • 发表日期：

    2025年3月22日

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  • Waleed Aziz

摘要

The linearizability problem of a singular point at the origin for a family of a quadratic four dimensional Lotka–Volterra system is investigated for (1:−1:1:1)-resonance. A complete set of necessary and sufficient conditions is obtained on the parameters that guarantee the existence of a coordinate change which transforms the system to a linear system. Several methods and techniques are used to prove the sufficiency of the obtained conditions. Mainly the Darboux linearization method and the linearizability of a singular point at the origin in the Poincaré domain are used in three variables with power-series arguments in the fourth variable. In some cases, if the system is completely integrable and there is a substitution that linearizes one of the equations, then the system is linearzable.