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

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

.