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 [math]\displaystyle{ \mathbf{\dot{x}} = f ( \mathbf{x} ) }[/math], [math]\displaystyle{ \mathbf{x} = [ x_1 \, x_2]^{\mathsf{T}} \in \mathbb{R}^2 }[/math], where [math]\displaystyle{ f(\mathbf{x}) = \begin{bmatrix} f^1 (\mathbf{x}) & f^2 (\mathbf{x}) \end{bmatrix}^{\mathsf{T}} }[/math], for which [math]\displaystyle{ \mathbf{x}^\ast = \mathbf{0} }[/math] is an equilibrium point, is uniformly globally asymptotically stable if:

(a) the trace of the Jacobian matrix is negative, [math]\displaystyle{ \operatorname{tr} \mathbf{J}_f (\mathbf{x}) \lt 0 }[/math] for all [math]\displaystyle{ \mathbf{x} \in \mathbb{R}^2 }[/math],

(b) the Jacobian determinant is positive, [math]\displaystyle{ \left| \mathbf{J}_{f} (\mathbf{x}) \right| \gt 0 }[/math] for all [math]\displaystyle{ \mathbf{x} \in \mathbb{R}^{2} }[/math], and

(c) the system is coupled everywhere with either

[math]\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. }[/math]