# Hopf Bifurcation Theorem

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.

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 $$0.69 and negative for $$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 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!

• Just few comments: 1) If you've already found equilibrium curves for your system, would you mind sharing it with others to ease answering the question? It's not easy looking nonlinear equation to solve, at first glance at least. 2) PDF from your link has some missing notation, especially concerning where w.r.t. critical value of parameter limit cycle exists. Can you check your calculations against Scholarpedia? – Evgeny Mar 24 at 9:25
• @Evgeny I've just added the nullclines+vector field. As for the Scholarpedia, there are two important values: the derivative of the real part of the eigenvalue and 1st coefficient of Lyapunov, just like in the pdf I've posted. The thing is that I'm not sure how the Lyapunov coefficient is equal to that expression in terms of third and second derivatives of $f$ and $g$ (I don't know how to evaluate it in the way Scholarpedia writes it). – AJHC Mar 24 at 11:07
• 2) Just use the last formula in this section. You can compute derivatives of RHS symbolically and just plug equilibria coordinates. 1) I mean, everyone who would want to analyze bifurcations here would like to know equilibria coordinates depending on $K$. Have you solved the equation for them or have coordinates only numerically? – Evgeny Mar 24 at 18:30
• I almost agree with your conclusions except for description of changes in phase portrait. For me the description is the following. A stable focus loses stability at $K \approx 0.69$, becomes an unstable focus and spawns a stable limit cycle. This limit cycle exists when $0.69 < K < 0.84$, but at $K \approx 0.84$ it stucks back to the equilibrium, ceases to exist and unstable focus becomes stable again. I can't memorize which one is subcritical and which is supercritical, but this seems to be consistent with eigenvalue derivatives and Lyapunov values. – Evgeny Mar 25 at 18:19
• 1) I'm talking about attractors. Saddle is not an attractor. When you have $K > 0.69$ pick a point near the stable limit cycle. Memorize that point and check if you go to the limit cycle from this point (or its vicinity) when $K < 0.69$. 2) I'm suggesting that if you see unstable small-amplitude cycle before critical value and large-amplitude stable limit cycle after the critical value, it doesn't contradict theorem about Andronov-Hopf bifurcation. It's perfectly compatible with that since theorem deals with small parameter values and small neighbourhood of focus. – Evgeny May 3 at 23:03