Coming back on the system I already mentioned in another post, this time I am working on some bifurcation analysis of a 2D System.
The system is defined by the following equations. I am assuming $\tau_a >1$ to be kept fixed.
\begin{equation} \begin{aligned} \dot d_{1} &= - d_1 - e_1 + \varepsilon f(d_1) \\ \tau_a \dot{e}_{1}&=g_a f(d_1) - e_{1}\\ f(d_1) &= \frac{e^{d_1}-1}{e^{d_1}+1} \end{aligned} \label{eq:2D_sync} \end{equation}
In particular, one can show that a Hopf bifurcation occurs when $\epsilon>\epsilon^*,\ \epsilon^*=2(1 + \frac{1}{\tau_a})$. I mentioning it here for the sake of completeness, although it is not the main problem I am trying to solve.
In orange are plotted some trajectories of the system forward in time
In blue are plotted some trajectories of the system backward in time
In red are plotted the nullclines
After Hopf bifurcation:
By increasing the parameter $\epsilon$, the size of the limit cycle increases until two fixed points emerges "from inside" the limit cycle. In the picture below you can see the intersection of the two nullclines.
By further increasing $\epsilon$ the limit cycle collapses on the fixed points that become stable.
Zooming on the new stable fixed points.
It looks again as an Hopf bifurcation but I did not find anything to classify it properly.
Do you have any suggestion on some more rigorous analysis/example?
UPDATE
Numerically I computed the value of $\epsilon^{o}=13.12$ at which the off-origin equilibria bifurcates. It is a subcritical Hopf bifurcation. In the figure below I plot the system's evolution with $\epsilon=13.23$.
and just before the catastrophe ($\epsilon=13.246$) (the two blue limit cycles also merge, it seems). In the plot below, only two trajectories are plotted (forward in time in orange, backward in time in blue). The initial transient has been removed as well.
I would say that the unstable (in blue) and stable (in orange) limit cycles collides and annihilate but I think I need something more rigorous or a similar system to motivate this intuitive consideration.