FINDING EQUILIBRIUM POINTS FOR NONLINEAR SYSTEM

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Ahmed
Ahmed on 12 Feb 2014
Commented: Mohammad Bader on 4 Jul 2020
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
  1 Comment
Mohammad Bader
Mohammad Bader on 4 Jul 2020

Can you help me to find the equilibriun point of this system X_n+1 = A + ( y_n ÷ x_n-k ) Y_n+1 = B + ( x_n ÷ y_n-k )

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Accepted Answer

Roger Stafford
Roger Stafford on 12 Feb 2014
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.
  4 Comments
Roger Stafford
Roger Stafford on 13 Feb 2014
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

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