A note on local integrability of differential systems

Abstract

For an $n$--dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(|x|^2)$, the Poincar\'e 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.Comment: 1

Analytic integrability of generalized 3-dimensional chaotic systems

“…Theorem 7 . [ 36 ] Consider the polynomial differential system ( 2 ). Let’s assume that λ 1 = 0, λ 2 and λ 3 are eigenvalues of Jacobian matrix at origin.…”

Analytic integrability around a nilpotent singularity

“…The most studied resonant saddles are resonant saddles of Lotka-Volterra systems, see [10,11,18,20,21]. The local analytic integrability problem for some particular differential systems in the plane and for n-dimensional differential systems is studied in [12,22,27,29]. We now introduce some notation in order to present the main results of this work.…”

A survey on local integrability and its regularity

“…After this result was published in 2003, a long time has passed in before we could determine whether the formal first integral in statement (b) of Theorem 6(b) could be replaced by some first integrals with suitable regularity. Zhang [30] in 2017 answered this problem under suitable conditions on the nonresonant eigenvalues of A.…”

“…Theorem 7 has a C ∞ version, see Theorem 2 of Zhang [30]. But when the eigenvalues µ have both positive real parts and negative real part, as shown in Theorem 3(b 2 ) of [30], there exist systems of form (2) which have no analytic first integrals defined in a neighborhood of the origin. Then one has to ask: under this last condition, does system (2) have C ∞ first integrals in a neighborhood of the origin.…”