Consider the nonlinear dynamical system $(1)$ :
$x' = y(1 + x−y^2)$,
$y' = x(1 + y−x^2)$, where $(x,y)\in\mathbb{R}^2$.
(i) Determine the equilibrium points of $(1)$
(ii) Classify the equilibrium points found in part (i)
(iii) Suppose that the equations model an experimental situation such that $x≥0$, $y≥0$
($x$, $y$ could for example be related to the concentrations of chemical species in a chemical reaction). Do periodic orbits exist?
I got:
(i) $(0,0)$, $(0,1)$, $(0,-1)$, $(1,0)$, $(-1,0)$, $(\frac{1+\sqrt5}2,\frac{1+\sqrt5}2)$, $(\frac{1-\sqrt5}2,\frac{1-\sqrt5}2)$
(I am pretty sure these are correct and I haven't missed out any)
(ii) Saddle point, Unstable Spiral, cannot be classified, Unstable Spiral, cannot be classified, Saddle point, Stable Spiral respectively to the order I wrote the equilibrium points.
How do you do (iii)????
To repeat a comment about (iii), which does not seem to have been taken into consideration: for most initial conditions in the positive quadrant, the solution ends up out of the positive quadrant (the exceptional initial conditions are the fixed points $(1,0)$ and $(0,1)$ and the diagonal $x=y$). Hence this system is ill-suited to model the evolution of a pair of populations. Proof:
$\qquad\qquad\qquad\qquad$
$$\texttt{streamplot[{y(1+x−y^2),x(1+y−x^2)},{x,-0.5,3},{y,-0.5,3}]}$$