Olech theorem explained

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,^[1] based on joint work with Philip Hartman.^[2]

Theorem

The differential equations

• x = f ( x ) , x = [ x _{1} x ^{T} _{2 ]} \in R ^{2} , where f( x ) = \begin{bmatrix} f ^{1} ( x ) & f ^{2} ( x ) \end{bmatrix} ^{T} , for which x ^{\ast} = 0 is an equilibrium point, is uniformly globally asymptotically stable if:

(a) the trace of the Jacobian matrix is negative,

\operatorname{tr} J _{f} ( x ) < 0 for all x \in R ^{2} ,

(b) the Jacobian determinant is positive,

\left| J _{f} ( x ) \right| > 0 for all x \in R ^{2} , and

(c) the system is coupled everywhere with either

(\partial f ^{1})/(\partial x _{1}) (\partial f ^{2})/(\partial x _{2}) ≠ 0 , or (\partial f ^{1})/(\partial x _{2}) (\partial f ^{2})/(\partial x _{1}) ≠ 0 forall x \in R ^{2.}

Notes and References

1. Czesław . Olech . On the Global Stability of an Autonomous System on the Plane . Contributions to Differential Equations . 1 . 3 . 389–400 . 1963 . 0589-5839 .
2. Philip . Hartman . Czesław . Olech . On Global Asymptotic Stability of Solutions of Differential Equations . . 104 . 1 . 1962 . 154–178 . 1993939 .