Olech theorem

In dynamical systems theory, the Olech theorem establishes sufficient conditions for global asymptotic stability of a two-equation system of non-linear differential equations. The result was established by Czesław Olech in 1963, based on joint work with Philip Hartman.

Theorem

The differential equations $\dot{𝐱}=f(𝐱)$, $𝐱=[x_1{x}_2{]}^{𝖳}\in {ℝ}^2$, where $f(𝐱)={\left[\begin{array}{cc}{f}^1(𝐱)& f^2(𝐱)\end{array}\right]}^{𝖳}$, for which ${𝐱}^{\ast }=\mathrm{𝟎}$ is an equilibrium point, is uniformly globally asymptotically stable if:

(a) the trace of the Jacobian matrix is negative, $\mathrm{tr}{𝐉}_f(𝐱)<0$ for all $𝐱\in {ℝ}^2$,

(b) the Jacobian determinant is positive, $|{𝐉}_f(𝐱)|>0$ for all $𝐱\in {ℝ}^2$, and

(c) the system is coupled everywhere with either

$\frac{\partial f^1}{\partial x_1}\frac{\partial f^2}{\partial x_2}\ne 0,\text{ or }\frac{\partial f^1}{\partial x_2}\frac{\partial f^2}{\partial x_1}\ne 0\text{ for all }𝐱\in {ℝ}^2.$