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Volume 7, Number 3, 2017, Pages 872-883                                                                DOI:10.11948/2017055
Three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras
Nail H. Ibragimov and Aliya A. Gainetdinova
Keywords:Time dependent dynamical system, nonlinear superposition of solutions, Vessiot-Guldberg-Lie algebra $L_4$, standard forms of $L_4.$
      Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras $L_4.$ The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to \textit{standard forms}. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures.
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