A criterion for the stability of motion of nonlinear system (Q803367)

The author considers the nonlinear autonomous system (1) \(\dot x=f(x)\), \(f(0)=0\). He assumes that the Jacobian matrix of (1), \(J(x)=(\partial f_ i(x)/\partial x_ j)\), can be represented in the form \(J(x)=D(S(x)- C(x))\), where D is a real diagonal matrix, S(x) is a skew-symmetric matrix and C(x) is a positive definite symmetric matrix. He shows that then a necessary and sufficient condition for the equilibrium state of system (1) to be asymptotically stable is that all diagonal elements of D are positive. This result includes as a special case \((D=I)\) the well- known stability theorem of N. N. Krasovskij.