FINDING EQUILIBRIUM POINTS FOR NONLINEAR SYSTEM

I have this system dx1=(x1(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x2 dx2=x2(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x1,

I want to find the equilibrium points for this system in matlab

Answer 1

[Roger Stafford]

I assume equilibrium occurs when dx1 and dx2 are equal to zero. You don't need matlab here. This one is easy to solve by hand. Multiply the first equation by x2 and the second by x1, then subtract them. You get

 x1^2 = x2^2

which gives two possibilities, either x1 = x2 or x1 = -x2.

Case 1: x1 = x2

Substitute x1 for x2 in the first equation to get:

 4*x1^5-2*x1^3-(u-1)*x1 = 0

The five roots of this are easy to solve for explicitly. Either x1 = 0 or we can solve the quadratic equation in x1^2

 4*(x1^2)^2-2*(x1^2)-(u-1) = 0

which gives

 x1^2 = (1+/-sqrt(4*u-3))/4
 x1 = +/-sqrt((1+/-sqrt(4*u-3))/4)

These are the five possible roots for the case x1 = x2.

Case 2: x1 = -x2

This is very similar to the solution in case 1.

[Ahmed]

I think there will be just 4 possible roots not 5?

[Roger Stafford]

There should be nine roots altogether including the (0,0) root, but many of these will probably be complex-valued. It all depends on the value of 'u'.

[Ahmed]

when I subtract them. I will get

 x1^2 = -x2^2

Is there a big difference?

[Roger Stafford]

No, you made a mistake in your algebra, Saeed. You should get x1^2 = x2^2.

Original equations:

 x1(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x2 = 0
 x2(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x1 = 0

Multiply the first equation by x2 and the second one by x1:

 x2*x1(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x2*x2 = 0
 x1*x2(u+(x1^2+x2^2)-(x1^2+x2^2)^2)-x1*x1 = 0

Now subtract the second from the first:

 -x2^2+x1^2 = 0
 x1^2 = x2^2

[praveen]

how to solve using matlab