I have a 2D dynamical system of the form
\begin{cases} \dot{x}=f(x,y,K) \\[1ex] \dot{y}=g(x,y,K) \end{cases}
where $K$ is a free parameter (later I can write the system here). I've found two Hopf bifurcations at approximately $K=0.69$ and $K=0.84$. In between these two values, there is a clear stable limit cycle, and for $K<0.69$ and $K>0.84$ the oscilations die after some time $t$.
Furthermore, after computing the eigenvalues, I can see that the real part of the eigenvalue is negative for $K<0.69$ and $K>0.84$, and it is positive for $0.69<K<0.81$.
With all this, I think I can say that there is a supercritical bifurcation at $K=0.69$ because there are stable oscillations for $K>0.69$. As for $K=0.84$, I don't think we can say anything unless we go to Hopf theorem in here, and compute the derivative of the real part of the eigenvalue and the $a$ value which I think is the 1st Lyapunov coefficient (although I have no idea how the complicated expression for this coefficient turns out to be equal to this...?)
Now, the problem is that I've computed these two things at $K=0.69$ and $K=0.84$ and they are both positive in the two cases. So, according to the theorem this would mean that periodic solutions exist for $K<0.69$ and $K<0.84$. The second case is correct, but the first is not.
Moreover, the theorem states that the fixed point is stable for $K<0.69$ and $K<0.84$, and unstable for $K>0.84$, which is clearly not the case as it is easy to see from the signal of the real part of the eigenvalues!
So, what's happening here?
EDIT: The system is:
\begin{cases} \dot{x}=-\frac{8}{3}(-0.99e^{-0.01y}+1)(x^2-x)+Kxy\\[1ex] \dot{y}=\frac{4}{x^{3/2}}\left(-y+1+\frac{3}{2}\frac{y}{x}(x-1)-0.1\frac{x-1}{y+1}-K\right) \end{cases}
and the nullclines+vector field for the case $K=0.69$ is:
EDIT2: After discussing in the comments with Evgeny, I realized that I was evaluating the derivative of the real part of the egeinvalue incorrectly. Indeed, by simply analysing that it changes from negative to positive at $K=0.69$ and from positive to negative at $K=0.84$, we can say that its derivative at these points is positive and negative, respectively.
Furthermore, because the first Lyapunov coefficient is positive in both cases, we can conclude through Hopf theorem that there is a subcritical bifurcation at $K=0.69$ and a supercritical at $K=0.84$. So this means that the stable periodic oscillations that I mentioned I can clearly see between $K=0.69$ and $K=0.84$ are due to the supercritical bifucartion at $K=0.84$, and not the bifurcation at $K=0.69$.
EDIT3: I've just realized that the conclusions I've drawn in EDIT2 are not consistent with Hopf theorem! And so my question still remains:
The real part of the eigenvalue is positive for $0.69<K<0.84$ and negative for $K<0.69$ and for $K>0.84$. For $K<0.69$ I get this situation:
which I can't understand if it's a no limit-cycle situation or an unstable situation. For $0.69<K<0.84$ I have this situation:
which is clearly a stable limit cycle! Finally for $K>0.84$ I get a situation similar to $K<0.69$. With all this, it seems that the first Lyapunoc coefficient should be negative! But when I do the calculation I get a positive one!