I have the following ODE system:
and the questions:
(a) Find all the critical points, and investigate their (linear) stability, as the parameters β and δ are varied.
(b) Set β = 2. Note that if δ = 0, the origin is a critical point, and find its linear stability. Investigate the bifurcation as δ crosses the value 0
(c) Investigate the presence of limit cycles both analytically and numerically
At this stage i haven't done anything this complicated and I'm a bit confused by it.
For part (a) I solved through to get $u=\frac {\beta(\beta-1)\pm\sqrt{(1-\beta)^2 -4\delta}}{2}$ and $v=\frac {(\beta-1)\pm\sqrt{(1-\beta)^2 -4\delta}}{2}$. I then took the Jacobian matrix $$\begin{pmatrix} -1 & \beta \\ -1 & 2v+1\\ \end{pmatrix} $$ to try and understand whats happening at the critical points but I don't really understand whats happening when v is put in.
For part (b) I feel like i need to follow on from part a but again I'm not sure if i need to transform the system to put it into Normal form (which is the hint that was given)
And part (c) again feels like i probably need some information from the previous parts.
To understand stability in part (a), find the eigenvalues of the Jacobian. This will tell you what kind of fixed point it is (saddle, node, spiral etc.)
For part (b), set $b=0$ as instructed. Then, analyse the eigenvalues and see how the eigenvalues change as $\delta $ crosses $0$. This will tell you the type of bifurcation occurring.
For part (c), check when does the fixed point become repelling, and whether far-off trajectories are diverging or converging towards the fixed point. If they are converging, they must converge somewhere, which is not the fixed point (as it is repelling). Thus, as it is a 2D system, they must converge to a limit cycle (by the Poincare-Bendixon Theorem)