Local Integrability and Linearizability for Three Dimensional Lotka–Volterra Cubic Systems

• Aween Karim ^[1] ; Waleed Aziz ^[1] ; Azad Amen ^[2]
  1. [1] Salahaddin University-Erbil

     Salahaddin University-Erbil

     Irak

     
  2. [2] Duhok University, Salahaddin University-Erbil, Soran University,
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 1, 2025
• Idioma: inglés
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• Resumen

  • 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.

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