Existence of inverse integrating factors and Lie symmetries for degenerate planar centers
Copyright policy )Existence of inverse integrating factors and Lie symmetries for degenerate planar centers
In this paper, the authors prove that every analytic planar vector field \(X\) with an isolated (possibly degenerate) center singularity at \(0\in\mathbb{R}^2\) admits a smooth inverse integrating factor (IIF) on a neighbourhood \(B\) of \(0\) that is positive in \(B-\{0\}\) and flat at the origin. This result is shown to be optimal in the sense that the existence of an analytic IIF does not fully characterise the analytic center, as there are analytic centers for which every smooth IIF on a neighbourhood of the center is flat at the center (e.g., certain polynomial planar vector fields with a degenerate center). This result solves an open problem posed in [\textit{J. Giné} and \textit{S. Maza}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 2, 695--704 (2011; Zbl 1219.34049)] and complements the achievements described in [\textit{L. Mazzi} and \textit{M. Sabatini}, J. Differ. Equations 76, No. 2, 222--237 (1988; Zbl 0667.34036)]. Subsequently, it is proved that a sufficient condition for the existence of an analytic IIF for \(X\) on a neighbourhood of the origin is the existence of an analytic non-trivial first integral on such a neighbourhood. Additionally, it is remarked that this result remains valid even when the isolated singularity of \(X\) is not a center. Finally, it is shown that \(X\) also possesses a smooth Lie symmetry in a neighbourhood \(B\) of \(0\) that is nonvanishing and orthogonal to \(X\) on \(B-\{0\}\) and flat at the origin, solving an open problem stated in [\textit{J. Giné}, ``Lie symmetries and the center problem'', J. Appl. Anal. Comp. 1, No. 4, 487--496 (2011)].
Geometric methods in ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, inverse integrating factor, Planar vector field, Lie symmetry, first integral, Degenerate center, degenerate center, Inverse integrating factor, First integral, planar vector field, Symmetries, invariants of ordinary differential equations, Analysis
Geometric methods in ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, inverse integrating factor, Planar vector field, Lie symmetry, first integral, Degenerate center, degenerate center, Inverse integrating factor, First integral, planar vector field, Symmetries, invariants of ordinary differential equations, Analysis
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