MATLAB: Critical points for system of 3 first order differential equations

non-linear system

Hi all,

I am trying to solve a system of non-linear ODE's for the critical points. I then need to determine the stability of the points.

My system is as follows:

u'1 = u1 + 4*u2 – u1*u2

u2' = 9*u1 + 4*u2 – u2*u3

u3' = 2*u1^2 + 9*u2^2 -89

It was recommended to me to try Newton's method to find the critical points…but I am unsure how to do so for 3 equations.

The methods that I have learned up until this point won't work, considering the unknowns. I'm not asking for a solution…simply a nudge in the right direction.

Thank you

Best Answer

What would you define a critical point to be? Perhaps one where u'1=u'2=u'3=0?

How would you identify that set of points? Perhaps as solutions to the corresponding nonlinear system? It would seem the derivatives go away there. ;-)

Can Newton's method be applied to such a problem? Hint: do some reading about Newton-Raphson.

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MATLAB: I want to find the determinant for 5.5 matrix

>> syms s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
>> M = [0 -s1 -s2 -s3 -s4; s1 0 -s5 -s6 -s4; s2 s5 0 -s8 -s9; s3 s6 s8 0 -s10; s4 s7 s9 s10 0];
>> det(M)
ans =
s1*s4^2*s8^2 - s1*s4*s7*s8^2 - s2*s4^2*s6*s8 + s3*s4^2*s5*s8 - s3^2*s4*s5*s9 - s2^2*s4*s6*s10 + s3^2*s5*s7*s9 + s2^2*s6*s7*s10 + s2*s3*s4*s5*s10 + s2*s3*s4*s6*s9 + s1*s2*s4*s8*s10 - s1*s3*s4*s8*s9 - s2*s3*s5*s7*s10 - s2*s3*s6*s7*s9 + s2*s4*s6*s7*s8 - s3*s4*s5*s7*s8 - s1*s2*s7*s8*s10 + s1*s3*s7*s8*s9
>> simplify(ans)
ans =
(s4 - s7)*(s3*s5 - s2*s6 + s1*s8)*(s2*s10 - s3*s9 + s4*s8)

MATLAB: Matlab solve indeterminate polynomial equations numerically

Finally, I find out the answer, it seems matlab will do some approximations, the true system is

eqstrue=[ u6*(3*u5 - 1) - u4 - u1 - 3*u2*u3 - 3*u5*u6 + 3*u3*(u2 - 1) + 1/24, u2^2*u3*(u2 - 1) - 2*u5^3*u6 - u2*u3*(u2 - 1)^2 - (2*u4)/27 - u5*u6*(3*u5 - 1)^2 + u5^2*u6*(3*u5 - 1) + 1/360, 1/1260 - 2*u5^4*u6 - 2*u5^2*u6*(3*u5 - 1)^2 - 2*u2^2*u3*(u2 - 1)^2 - (2*u4)/81, u5^2*u6*(3*u5 - 1)^3 - 3*u5^5*u6 - u4/81 - 4*u5^3*u6*(3*u5 - 1)^2 + 4*u5^4*u6*(3*u5 - 1) + 1/2880, 9*u5^4*u6*(3*u5 - 1) - 3*u5^3*u6*(3*u5 - 1)^2 + 1/6720, u5^3*u6*(3*u5 - 1)^3 - 6*u5^4*u6*(3*u5 - 1)^2 - 3*u5^4*u6*(3*u5 - 1)^3 - 6*u5^5*u6*(3*u5 - 1)^2 + 9*u5^5*u6*(3*u5 - 1) - 3*u5^6*u6*(3*u5 - 1) + 1/21600, u5^3*u6*(3*u5 - 1)^3 - 6*u5^4*u6*(3*u5 - 1)^2 + u5^4*u6*(3*u5 - 1)^3 + 2*u5^5*u6*(3*u5 - 1)^2 + 9*u5^5*u6*(3*u5 - 1) + u5^6*u6*(3*u5 - 1) + 11/226800]

after I make

a2 = 1; b2 = -1; b3 = 1;

matlab solves the approximatied system

eqs=[ u6*(3*u5 - 1) - u4 - u1 - 3*u2*u3 - 3*u5*u6 + 3*u3*(u2 - 1) + 1/24, u2^2*u3*(u2 - 1) - 2*u5^3*u6 - u2*u3*(u2 - 1)^2 - (2*u4)/27 - u5*u6*(3*u5 - 1)^2 + u5^2*u6*(3*u5 - 1) + 1/360, 1/1260 - 2*u5^4*u6 - 2*u5^2*u6*(3*u5 - 1)^2 - 2*u2^2*u3*(u2 - 1)^2 - (2*u4)/81, u5^2*u6*(3*u5 - 1)^3 - 3*u5^5*u6 - u4/81 - 4*u5^3*u6*(3*u5 - 1)^2 + 4*u5^4*u6*(3*u5 - 1) + 1/2880, 9*u5^4*u6*(3*u5 - 1) - 3*u5^3*u6*(3*u5 - 1)^2 + 1/6720, u5^3*u6*(3*u5 - 1)^3 - 6*u5^4*u6*(3*u5 - 1)^2 - 3*u5^4*u6*(3*u5 - 1)^3 - 6*u5^5*u6*(3*u5 - 1)^2 + 9*u5^5*u6*(3*u5 - 1) - 3*u5^6*u6*(3*u5 - 1) + 34433922270924495617/743772721051969121157120, u5^3*u6*(3*u5 - 1)^3 - 6*u5^4*u6*(3*u5 - 1)^2 + u5^4*u6*(3*u5 - 1)^3 + 2*u5^5*u6*(3*u5 - 1)^2 + 9*u5^5*u6*(3*u5 - 1) + u5^6*u6*(3*u5 - 1) + 37713343439583972437/743772721051969121157120];

so it gives me an answer, I don't know why it does this. In the last equation.

vpa(11/226800 - 37713343439583972437/743772721051969121157120)
ans = -0.0000022045855379188659798651850341144609

It's a bad approximation.

Best Answer

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MATLAB: I want to find the determinant for 5.5 matrix

MATLAB: Matlab solve indeterminate polynomial equations numerically

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