A choice of Lyapunov function for this 2D system?

Question

I am thinking of a choice of a suitable Lyapunov function$V(x_{1},x_{2})$ which can make the system stable around the fixed point $x_{1}=1,x_{2}=1$

$\dot{x_{1}} = x_{1}x_{2} - x_{1}^2 $

$\dot{x_{2}} = x_{2} - x_{1}x_{2} + 2 -2x_{2}^2$

I thought of using $V(x_{1},x_{2}) = a(x_{1}-1)^2 + b(x_{2}-1)^2$, but then I am unable to show the $\frac{dV}{dt} <0$ but still we have $V(1,1) =0$, which I asked here - Determining $a$ and $b$ such that the expression is negative? and it seems I cannot take such a $V$? any help with the expression of $V(x_{1},x_{2})$?

Answer 1

You only need that $\dot V<0$ close to the sink $(1,1)$, so check the expression you originally tried, $V=a(x-1)^2+b(y-1)^2$ with \begin{align} \dot V&=2a(x−1)(xy−x^2)+2b(y−1)(y−xy+2−2y^2)\\ &=2a(x−1)x(y-x)+2b(y−1)(y(1-x)+2(1+y)(1-y)\\ &=-2ax(x-1)^2-2b(1+y)(y-1)^2~+~(2ax-2by)(x-1)(y-1) \end{align} Thus for $a=b=1$ you get for $(x,y)\approx(1,1)$ that $$\dot V\approx-2(x-1)^2-4(y-1)^2$$ which is negative. The remainder is of size $V^{3/2}$, thus smaller than the quadratic contributions close to $(1,1)$.

In some rough but exact estimate, \begin{align} \dot V&=-2(x-1)^2-4(y-1)^2~-~2(x-1)^3-2(y-1)^3+2(x-1)^2(y-1)-2(x-1)(y-1)^2\\ &\le-2V+8V^{3/2} \end{align} which is negative for $4V^{1/2}<1$ or $V<\frac1{16}$. Using better estimates that combine some of the third order terms one might get larger bounds, \begin{align} \dot V&=-2(x-1)^2-4(y-1)^2~-~2(x-1)^3-2(y-1)^3+2(x-1)^2(y-1)-2(x-1)(y-1)^2\\ &\le-2V+2\Bigl[(x-1)^2+(y-1)^2\Bigr]\bigl(|x-1|+|y-1|\bigr)\\ &\le-2V+2\sqrt2V^{3/2}, \end{align} which means that the admissible region is $V<\frac12$.

Answer 2

Using $V(x,y) = (x-1)^2+(y-1)^2$

First, a Lyapunov function, doesn't make a system stable. It serves to certify stability.

Second, the dynamical system has four equilibrium points as can be depicted in the attached stream plot.

Third, the sought local lyapunov function characterizes a limited region around the equilibrium point $(1,1)$ which is an approximation to the attraction basin for $(1,1)$

Attached the MATHEMATICA script

V0 = (x - 1)^2 + (y - 1)^2; gr0 = StreamPlot[{x y - x^2, y - x y + 2 - 2 y^2}, {x, -3, 3}, {y, -3, 3}, Mesh -> 20]; sols = Solve[{x y - x^2 == 0, y - x y + 2 - 2 y^2 == 0}, {x, y}] gr1 = ContourPlot[V0, {x, 1 - 1, 1 + 1}, {y, 1 - 1, 1 + 1}, ContourShading -> None, Contours -> 30, RegionFunction -> Function[{x, y}, (x - 1)^2 + (y - 1)^2 < 1/2]]; gr = Table[Graphics[{Red, Disk[({x, y} /. sols[[k]]), 0.1]}], {k, Length[sols]}]; Show[gr0, gr1, gr]