Hello again. I have another question!
Consider x dot = 1-2*x+x^2*y and y dot = x - x^2 * y
Find the fixed point(s) and determine their stability. Sketch the phase plane.
My attempt: I set x dot = y dot = 0 and found that the only fixed point for the system is (1,1). Then I found the jacobian matrix (2x2 matrix) for the system as [-2+2*x*y x^2 ; 1-2*x*y -x^2]
Then Jacobian matrix J1 for my fixed point is [ 0 1; -1 -1]
Finding then the determinant of λ*Ι - J1 leads me to 2 complex eigenvalues λ1 = -0.5 + sqrt(3) / 2 i and λ2 = -0.5-sqrt(3)/2.
Also the trace of the matrix is <0 so our fixed point is a stable point and the phase plane is an incoming spiral to (1,1)
If someone can confirm my work! Thank you very much again!