A geometric proof of Malkin's stability theorem. (Q1411776)

The author introduces a short geometric proof of Malkin's stability theorem for nonlinear systems of differential equations using center manifold theory for flows \[ \dot{x}_{1}=Ax_{2}+p(x_{1},x_{2}),\quad\dot{x}_{2}=Bx_{2}+q(x_{1},x_{2}), \] where \(A,B\) are constant matrices, \(B\) is Hurwitz, \(p,q\) are analytic functions vanishing at \((0,0)\) together with all their first order partial derivatives, and \(q(x,0) = O(\|x\|^{n+1})\), \(n\geq2\) [see \textit{I. G. Malkin}, Theory of stability of motion. Second revised edition Izdat. Nauka, Moscow (1966; Zbl 0136.08502)]. If \(p(x,0) = O(\|x\|^{n})\) and the equilibrium \(x_{1} = 0\) of the reduced system \(\dot{x}_{1}=p(x_{1},0) \) is stable, asymptotically stable or unstable, then the equilibrium \((x_{1},x_{2}) = (0,0)\) of the system is stable, asymptotically stable or unstable, respectively.