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,{{cite journal |first=Czesław |last=Olech |title=On the Global Stability of an Autonomous System on the Plane |journal=Contributions to Differential Equations |volume=1 |issue=3 |pages=389–400 |year=1963 |issn=0589-5839 }} based on joint work with Philip Hartman.{{cite journal |first=Philip |last=Hartman |first2=Czesław |last2=Olech |title=On Global Asymptotic Stability of Solutions of Differential Equations |journal=Transactions of the American Mathematical Society |volume=104 |issue=1 |year=1962 |pages=154–178 |jstor=1993939 }}

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

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

References

Category:Theorems in dynamical systems

Category:Stability theory