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

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