Question

Alright here is a non-linear dynamics problem: Given {x'=x^2 +y} and {y' = x-y+a} where {a} is a parameter, find (a) all equilibrium points and compute the linearized equation for each (b) Describe the behavior of the linearized system for each equilibrium point (c) describe any bifurcations that occur.

Answer 1

(a) Equilibrium is when x' = y' = 0
=> x^2 + y = 0 and y = x + a
=> x^2 + x + a = 0

You can now solve for x and then y.

You tell me what's next

Answer 2

Well if I take a=0 then x=y and if x=y=-1 then (-1,-1) is an equilibrium point.

Answer 3

And another EP is (0,0)

Answer 4

I could try to take the Jacobian Matrix at the two equilibrium points...

Answer 5

How do I know if Bifurcations occur?

Answer 6

You can't make such assumptions about a, but consider three cases where the equation x^2 + x + a =0 has no, one or two roots.

As for bifurcation tests, this is where your text book or lecture notes are really important.

Answer 7

And how am i supposed to do a global transformation with:
x'=x^2 +y
y'=x-y
Would it be just:
u=qx^2 +y
v=wy+x
then:
u'=q2xx' +y'
v'=wy' +x'
And then if u'=u and v'=v then:
q2x(x^2 +y) +(x-y) = qx^2 +y and:
w(x-y ) +(x^2 +y) = wy+x
But now I need to solve for w and q and they need to be a number.... not variables...... I dont think its possible so i dont know why the transformation isnt working...