The center problem on a center manifold in \(\mathbb{R}^{3}\) (Q412737)

Consider the system of differential equations NEWLINE\[NEWLINE\begin{aligned} \dot u &=-v+ P(u,v,w),\\ \dot v &=u+ Q(u,v,w),\\ \dot w &=-\lambda w+ R(u,v,w),\end{aligned}NEWLINE\]NEWLINE where \(\lambda\) is a positive real number. It is shown that, for each fixed value of the non-zero real eigenvalue \(\lambda\), the set of such systems having a center on the local center manifold at the origin corresponds to a variety in the space of admissible coefficients.NEWLINENEWLINE The center-focus problem for the system NEWLINE\[NEWLINE\begin{aligned} \dot u &=-v+ au^2+ av^2+ cuw+ dvw,\\ \dot v &=u+ bu^2+ bv^2+ euw+ fvw,\\ \dot w &=-w+ Su^2+ Sv^2+ Tuw+ Uvw, \end{aligned}NEWLINE\]NEWLINE is solved in the following three casesNEWLINENEWLINE (1) \(S= 0\);NEWLINENEWLINE (2) \(a= b= c+ f=0\), \(S= 1\);NEWLINENEWLINE (3) \(d+ e= c= f= 0\), \(S= 1\).