Lyapunov–Malkin theorem

The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Gilevich Malkin) is a mathematical theorem detailing nonlinear stability of systems.

Theorem

In the system of differential equations,

${\dot {x}}=Ax+X(x,y),\quad {\dot {y}}=Y(x,y)\$

where, $x\in \mathbb {R} ^{m}$, $y\in \mathbb {R} ^{n}$, $A$ in an m × m matrix, and X(x, y), Y(x, y) represent higher order nonlinear terms. If all eigenvalues of the matrix $A$ have negative real parts, and X(x, y), Y(x, y) vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect to (x, y) and asymptotically stable with respect to  x. If a solution (x(t), y(t)) is close enough to the solution x = 0, y = 0, then

$\lim _{t\to \infty }x(t)=0,\quad \lim _{t\to \infty }y(t)=c.\$