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arXiv · 1712.09510

A note on local integrability of differential systems

Xiang Zhang
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Abstract

For an $n$--dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(|x|^2)$, the Poincaré nonintegrability theorem states that if the eigenvalues of $A$ are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when $A$ admits one zero eigenvalue and the other are non--resonant: for $n=2$ the system has an analytic first integral at the origin if and only if the origin is a non--isolated singular point; for $n>2$ the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of \emph{whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point} remains open.

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Xiang Zhang. 2017-12-27. A note on local integrability of differential systems. https://doi.org/10.1016/j.jde.2017.08.016

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