Local integrability of Poincaré-Dulac normal forms
This paper studies the analytic integrability of Poincaré-Dulac normal forms. The author considers dynamical systems of the form $\dot{x} = f(x)$ for $x\in\mathbb{C}^n$, where $n\ge 1$, $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ and $f(0) = 0$. Throughout it is assumed that $f$ has a power series expansion of the form $f(x)=\sum_{n=1}^{\infty} f^{i}$, where $f^i$ is a homogeneous vector field of degree $i$ for $i=1, 2,\dots$. If $A = Df(0)$, the matrix $A$ can be decomposed as $A = S + N$, where $S$ is semi-simple and $N$ is nilpotent. This leads to a decomposition $f^1=f^s(x)+f^n(x)$ where $f^s(x) = Sx$ and $f^n(x) = Nx$. The vector field \(f\) is said to be in Poincaré-Dulac normal form if the Lie bracket of $f^s$ and $f$, written $Df(x) f^s(x) - Df^s(x)f(x)$, vanishes for every $x\in\mathbb{C}^n$. \par The author reviews previous results for analytic integrability of the Poincaré-Dulac and Birkhoff normal forms [\textit{A. D. Bryuno}, Tr. Mosk. Mat. O.-va 25, 119--262 (1971; Zbl 0263.34003); Trans. Mosc. Math. Soc. 26, 199--239 (1974; Zbl 0283.34013); \textit{O. I. Bogoyavlenskij}, Commun. Math. Phys. 196, No. 1, 19--51 (1998; Zbl 0931.37028); \textit{Nguyen Tien Zung}, Math. Res. Lett. 9, No. 2--3, 217--228 (2002; Zbl 1019.34084); Ann. Math. (2) 161, No. 1, 141--156 (2005; Zbl 1076.37045)], and notes that there is a general transformation that produces a normal form for the original dynamical system but that transformation is not generally analytic. \par The main result of the paper is a sufficient condition for a dynamical system (having a form as described above) in Poincaré-Dulac normal form to be analytically integrable, and that condition is also necessary if certain technical criteria are met. The integrability conditions depend on the resonance degrees of the dynamical system. The author shows that systems with resonance degrees of 0 or 1 are integrable, but may not be if the resonance degrees are greater than 1. \par The author applies his theoretical results to a dynamical system with a codimension-two fold-Hopf bifurcation.
scientific article; zbMATH DE number 7048259
scientific article