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Integrability of complex planar systems with homogeneous nonlinearities. (English) Zbl 1337.34003

Consider the two-dimensional real system with quintic homogeneous nonlinearity \[ \begin{aligned} \dot{x} & =-y+a_{1}x^{5}+a_{2}x^{4} y+a_{3}x^{3} y^{2}+a_{4}x^{2} y^{3}+a_{5}x y^{4}+a_{6}y^{5}, \\ \dot{y} & =x+b_{1}x^{5}+b_{2}x^{4} y+b_{3}x^{3} y^{2}+b_{4}x^{2} y^{3}+b_{5}x y^{4}+b_{6}y^{5}. \end{aligned} \] The main interest is the local integrability of this system. The local integrability is equivalent to the existence of a center at the origin. The authors considered a more general family of complex systems \[ \begin{aligned} \dot{x} & =x+a_{1}x^{5}+a_{2}x^{4} y+a_{3}x^{3} y^{2}+a_{4}x^{2} y^{3}+a_{5}x y^{4}+a_{6}y^{5}, \\ \dot{y} & =-y+b_{1}x^{5}+b_{2}x^{4} y+b_{3}x^{3} y^{2}+b_{4}x^{2} y^{3}+b_{5}x y^{4}+b_{6}y^{5}, \end{aligned} \] where \(x\) and \(y\) are complex variables and \(a_{i},b_{i}\) are complex parameters. From the conditions of integrability of this complex system, one can derive conditions for integrability of the real system.
The authors present sufficient and necessary conditions for local integrability of the complex system when \(a_{6}=0\). The necessary conditions are obtained by computing the focal values. The sufficiency of these conditions are proved using the idea reversibility.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Software:

SINGULAR; primdec
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References:

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