Lyapunov–Malkin theorem

The Lyapunov–Malkin theorem is a mathematical theorem detailing stability of nonlinear systems.

Theorem

In the system of differential equations,
where and are components of the system state, is a matrix that represents the linear dynamics of, and and represent higher-order nonlinear terms. If all eigenvalues of the matrix have negative real parts, and X, Y vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect to and asymptotically stable with respect to x. If a solution, y) is close enough to the solution x = 0, y = 0, then

Example

Consider the vector field given by
In this case, A = -1 and X = Y = 0 for all y, so this system satisfy the hypothesis of Lyapunov-Malkin theorem.
The figure below shows a plot of this vector field along with some trajectories that pass near. As expected by the theorem, it can be seen that trajectories in the neighborhood of converges to a point in the form.