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Integrability and Linearizability of Three Dimensional Vector Fields

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Abstract

This paper deals with the integrability and linearizability problem of three dimensional systems

$$\begin{aligned} \dot{x}&= x(1 +ax+by+cz),\\ \dot{y}&= -y+ dx^2 +exy + fxz+gyz+hy^2+kz^2,\\ \dot{z}&= z(1 + \ell x + my +pz). \end{aligned}$$

More precisely, we give a complete set of necessary conditions for integrability and linearizability and then prove their sufficiency using Darboux method and Darboux inverse Jacobi multiplier, power series argument and a solution of a Riccati equation.

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Acknowledgments

I would like to thank Colin Christopher for helpful discussions during the preparation of this work.

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  1. Department of Mathematics, College of Science, Salahaddin University, Arbil, Iraq

    Waleed Aziz

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  1. Waleed Aziz
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Correspondence to Waleed Aziz.

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Aziz, W. Integrability and Linearizability of Three Dimensional Vector Fields. Qual. Theory Dyn. Syst. 13, 197–213 (2014). https://doi.org/10.1007/s12346-014-0113-0

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  • DOI: https://doi.org/10.1007/s12346-014-0113-0

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