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