Divergence of a real normalizing transformation
The following system is considered: (1) \(\dot z{}_ j=z_ j(\lambda_ j+f_ j(z_ 1,z_ 2))\), \(j=1,2\), where \(f_ j\) are analytic at zero \(z_ 1=0\), \(z_ 2=0\), \(f_ j(0,0)=0\) and \(\lambda_ 1=-\lambda_ 2\neq 0\). There exists a formal coordinate transformation (2) \(z_ j=w_ j(1+h_ j(w_ 1,w_ 2)), j=1,2\), reducing the system (1) to the normal form (3) \(\dot w_ j=w_ j(\lambda_ j+g_ j(w_ 1w_ 2)), j=1,2\), where \(h_ j\) and \(g_ j\) are formal power series of \(w_ 1,w_ 2\) without free members and \(g_ j\) depends only on the product \(w_ 1w_ 2\). Here only the case Re \(\lambda {}_ 1=Re \lambda_ 2\) is considered. The following theorem is proved: For each analytic normal form (3) satisfying some ''reality'' conditions, there exists an analytic system (1), formally equivalent to system (3) and for which any normalizing transformation (2) is divergent.
scientific article; zbMATH DE number 3865716
scientific article