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 $\dot{\mathbf{x}}=f(\mathbf{𝐱})$, $\mathbf{𝐱}=[x_1{x}_2{]}^{\mathsf{T}}\in {\mathbb{ℝ}}^2$, where $f(\mathbf{𝐱})={\left[\begin{array}{cc}{f}^1(\mathbf{𝐱})& f^2(\mathbf{𝐱})\end{array}\right]}^{\mathsf{T}}$, 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, $\mathrm{tr}{\mathbf{𝐉}}_f(\mathbf{𝐱})<0$ for all $\mathbf{𝐱}\in {\mathbb{ℝ}}^2$,

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

(c) the system is coupled everywhere with either

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