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The reversibility problem for quasi-homogeneous dynamical systems

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  • In this paper, we obtain necessary and sufficient conditions for the reversibility of a quasi-homogeneous $n$-dimensional system. As a consequence, we get necessary conditions for an arbitrary system to be orbital-reversible. Moreover, we give sufficient conditions for orbital-reversibility in terms of the existence of Lie symmetries of the vector field. The results obtained are conveniently adapted to the case of planar systems, where we give sufficient conditions for a degenerate planar vector field to have a center at the origin. We apply the results to some case studies. Namely, we consider a family of planar vector fields, where we determine centers which are not orbital-reversible. We also study some tridimensional systems, where nonlinear involutions are determined in the reversible situations.
    Mathematics Subject Classification: Primary: 34C14; Secondary: 34A26.

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