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 $$\mathbf{\dot{x}} = f ( \mathbf{x} )$$, $$\mathbf{x} = [ x_1 \, x_2]^{\mathsf{T}} \in \mathbb{R}^2$$, where $$f(\mathbf{x}) = \begin{bmatrix} f^1 (\mathbf{x}) & f^2 (\mathbf{x}) \end{bmatrix}^{\mathsf{T}}$$, for which $$\mathbf{x}^\ast = \mathbf{0}$$ is an equilibrium point, is uniformly globally asymptotically stable if:
 * (a) the trace of the Jacobian matrix is negative, $$\operatorname{tr} \mathbf{J}_f (\mathbf{x}) < 0$$ for all $$\mathbf{x} \in \mathbb{R}^2$$,
 * (b) the Jacobian determinant is positive, $$\left| \mathbf{J}_{f} (\mathbf{x}) \right| > 0$$ for all $$\mathbf{x} \in \mathbb{R}^{2}$$, and
 * (c) the system is coupled everywhere with either


 * $$\frac{\partial f^1}{\partial x_1} \frac{\partial f^2}{\partial x_2} \neq 0,

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