We will prove part of Theorem 9.3 .2 : If the critical point \((0,0)\) of the
almost linear system
$$
d x / d t=a_{11} x+a_{12} y+F_{1}(x, y), \quad d y / d t=a_{21} x+a_{22}
y+G_{1}(x, y)
$$
is an asymptotically stable critical point of the corresponding linear system
$$
d x / d t=a_{11} x+a_{12} y, \quad d y / d t=a_{21} x+a_{22} y
$$
then it is an asymptotically stable critical point of the almost linear system
(i). Problem 12 deals with the corresponding result for instability.
Consider the linear system (ii).
(a) Since \((0,0)\) is an asymptotically stable critical point, show that
\(a_{11}+a_{22}<0\) and \(\left.a_{11} a_{22}-a_{12} a_{21}>0 . \text { (See
Problem } 21 \text { of Section } 9.1 .\right)\)
(b) Construct a Liapunov function \(V(x, y)=A x^{2}+B x y+C y^{2}\) such that
\(V\) is positive definite and \(\hat{V}\) is negative definite. One way to ensure
that \(\hat{V}\) is negative definite is to choose \(A, B,\) and \(C\) so that
\(\hat{V}(x, y)=-x^{2}-y^{2} .\) Show that this leads to the result
$$
\begin{array}{l}{A=-\frac{a_{21}^{2}+a_{22}^{2}+\left(a_{11} a_{22}-a_{12}
a_{21}\right)}{2 \Delta}, \quad B=\frac{a_{12} a_{22}+a_{11} a_{21}}{\Delta}}
\\\ {C=-\frac{a_{11}^{2}+a_{12}^{2}+\left(a_{11} a_{22}-a_{12}
a_{21}\right)}{2 \Delta}}\end{array}
$$
where \(\Delta=\left(a_{11}+a_{22}\right)\left(a_{11} a_{22}-a_{12}
a_{21}\right)\)
(c) Using the result of part (a) show that \(A>0\) and then show (several steps
of algebra are required) that
$$
4 A
C-B^{2}=\frac{\left(a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+a_{22}^{2}\right)\left(a_{11}
a_{22}-a_{12} a_{21}\right)+2\left(a_{11} a_{22}-a_{12}
a_{21}\right)^{2}}{\Delta^{2}}>0
$$
Thus by Theorem 9.6.4, \(V\) is positive definite.
