Unstable equilibrium point for $f(x,y)=\left(\begin{array}{c}-y+xg(x,y)\\x+yh(x,y)\end{array}\right)$

Question

Let the Nonlinear System $$f(x,y)=\left(\begin{array}{c}-y+xg(x,y)\\x+yh(x,y)\end{array}\right)$$

$f$ and $g$ continuous, $f(0,0)>0$ and $g(0,0)>0.$

I want to prove that $(0,0)$ is an unstable equilibrium point.

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So I shlould find a Lyapunov function such that $V'(y)>0$ then $(0,0)$ is an unstable equilibrium point, but I don't know how find it, can you help me please?

In another Nonlinear systems what should I do to find Lyapunov functions?

Thank you

Answer 1

Considering the system

$$ \cases{ \dot x = -y + x g(x,y)\\ \dot y = x + y h(x,y) } $$

multiplying the first equation by $x$, the second by $y$ and adding we have

$$ \frac 12\frac{d}{dt}(x^2+y^2) = x^2 g(x,y) + y^2 h(x,y) $$

and inside a ball $\mathcal{B}$ containing the origin we have $g(x,y)>0,h(x,y)>0$ for $(x,y)\in\mathcal{B}$ hence

$$ \frac 12\frac{d}{dt}(x^2+y^2) = x^2 g_{min} + y^2 h_{min} \ge 0 \ \ \text{with equality when}\ \ \ (x=0, y=0) $$

here $g_{min} = \min_{(x,y)\in \mathcal{B}}g(x,y)$ and $h_{min} = \min_{(x,y)\in \mathcal{B}}h(x,y)$ so according to Lyapounov, the origin is not stable.