Kernel: SageMath 9.7

C34 Another Nonlinear System Example

Consider the following system of two nonlinear, autonomous differential equations:

x=y22yy=x2xy\qquad\begin{array}{lcl} x' &=& y^2 - 2y \\ y' &=& x^2 - x - y \end{array}

(a) Find the equilibria and their types via a linearization.

(b) Sketch a phase portrait on 4x,y4-4 \leq x, y \leq 4 with the nullclines and direction arrows in the regions of the plane created by the nullclines.

(c) Sketch a solution trajectory on the phase plane with x(0)=2x(0) = 2 and y(0)=2y(0) = -2, and sketch a plot of x(t)x(t) and y(t)y(t) versus tt on an open interval continaing t=0t = 0.

(d) Sketch a solution trajectory on the phase plane with x(0)=2x(0) = -2 and y(0)=1y(0) = 1, and sketch a plot of x(t)x(t) and y(t)y(t) versus tt on an open interval continaing t=0t = 0.

(e) Sketch additional representative solutions on the phase plane.

(f) Describe the long term behavior of solutions.